TOPICS ON MAX-STABLE PROCESSES AND THE CENTRAL LIMIT THEOREM by Yizao Wang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Statistics) in The University of Michigan 2012 Doctoral Committee: Associate Professor Stilian. A. Stoev, Chair Professor Tailen Hsing Professor Robert W. Keener Professor Roman Vershynin Professor Emeritus Michael B. Woodroofe
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TOPICS ON MAX-STABLE PROCESSES AND
THE CENTRAL LIMIT THEOREM
by
Yizao Wang
A dissertation submitted in partial fulfillmentof the requirements for the degree of
Doctor of Philosophy(Statistics)
in The University of Michigan2012
Doctoral Committee:
Associate Professor Stilian. A. Stoev, ChairProfessor Tailen HsingProfessor Robert W. KeenerProfessor Roman VershyninProfessor Emeritus Michael B. Woodroofe
ACKNOWLEDGEMENTS
First of all, I am indebted to my thesis advisor Professor Stilian A. Stoev for his
help and support since 2008. He has been a great mentor for me in my research
career. At the same time, he has also provided me many helps and advice in daily
life. This dissertation would not have been possible without him. In particular, the
first part of this dissertation is under his supervision.
Second, I am grateful to Professor Emeritus Michael Woodroofe. He sets up a
very high standard for scholars, and as a young researcher I am deeply influenced by
him in many aspects. The second part of this dissertation is under his supervision.
I would also like to thank Professor Yves Atchade, Professor Tailen Hsing, Pro-
fessor Bob Keener and Professor Parthanil Roy (from Michigan State University)
for many insightful and inspiring discussions on research. I also appreciate Profes-
sor Tailen Hsing, Professor Bob Keener, Professor Roman Vershynin and Professor
Michael Woodroofe for serving on my thesis committee.
I own many thanks to all the faculty members and students in the Department
of Statistics at the University of Michigan. I really enjoy my last five years as a
graduate student in Ann Arbor.
At last, I am greatly indebted to my parents for their unconditional support of
my pursue of academia career abroad during the past years. Without their support I
can achieve no success. I am also grateful to my wife, Fei Xu, for her companionship
5.1 Four samples from the conditional distribution of the discrete Smith model (seeSection 5.5), given the observed values (all equal to 5) at the locations marked bycrosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Prediction of a MARMA(3,0) process with φ1 = 0.7,φ2 = 0.5 and φ3 = 0.3, basedon the observation of the first 100 values of the process. . . . . . . . . . . . . . . . 77
5.3 Conditional medians (left) and 0.95-th conditional marginal quantiles (right). Eachcross indicates an observed location of the random field, with the observed value atright. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
v
LIST OF TABLES
Table
5.1 Means and standard deviations (in parentheses) of the running times (in seconds)for the decomposition of the hitting matrix H, based on 100 independent obser-vations X = A ⊙ Z, where A is an (n × p) matrix corresponding to a discretizedSmith model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Cumulative probabilities that the projection predictors correspond to at time 100+t, based on 1000 simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Coverage rates (CR) and the widths of the upper 95% confidence intervals at time100 + t, based on 1000 simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 78
vi
CHAPTER I
Introduction
This dissertation consists of results in two distinct areas of probability theory.
One is the extreme value theory, the other is the central limit theorem.
In the extreme value theory, the focus is on max-stable processes. Such processes
play an increasingly important role in characterizing and modeling extremal phenom-
ena in finance, environmental sciences and statistical mechanics. Several structural
and ergodic properties of max-stable processes are investigated via their spectral rep-
resentations. Besides, the conditional distributions of max-stable processes are also
studied, and a computationally efficient algorithm is developed. This algorithm has
many potential applications in prediction of extremal phenomena.
In the central limit theorem, the asymptotic normality for partial sums of sta-
tionary random fields is studied, with a focus on the projective conditions on the
dependence. Such conditions, easy to check for many stochastic processes and ran-
dom fields, have recently drawn many attentions for (one-dimensional) time series
models in statistics and econometrics. Here, the focus is on (high-dimensional) sta-
tionary random fields. In particular, a general central limit theorem for stationary
random fields and orthomartingales is established. The method is then extended to
establish the asymptotic normality for the kernel density estimator of linear random
1
2
fields.
Below are overviews of the following chapters of this dissertation.
1.1 Max-stable Processes
Max-stable processes arise in the limit of maxima of independent and identically
distributed processes. It is well known that all max-stable processes can be trans-
formed to α-Frechet processes. A random variable Y is α-Frechet with α > 0, if
P(Y ≤ y) = exp(−σαy−α), y > 0.
A stochastic process Ytt∈T is α-Frechet, if all its max-linear combinations in form
of maxi=1,...,n aiYti≡
n
i=1 aiYti, ai > 0, ti ∈ T, i = 1, . . . , n, n ∈ N are α-Frechet.
It is known since de Haan [23] that under mild regularity conditions, for every α-
Frechet process Ytt∈T , there exists a class of non-negative, Lα-integrable functions
ftt∈T ∈ Lα
+(S,BS, µ), such that
(1.1) P(Yt1 ≤ y1, . . . , Ytn≤ yn) = exp
−
S
n
i=1
fti(s)/yti
α
µ(ds).
Indeed, every such a process has an extremal integral representation as
(1.2) Ytt∈Td=
e
S
ft(s)M∨
α(ds)
t∈T
,
where ‘e
’ is the symbol of the extremal integral and M∨α
is an α-Frechet random
sup-measure (see Stoev and Taqqu [101]).
Preliminary results on max-stable processes can be found in Chapter II. Then,
starting with such representation results, structural properties of max-stable pro-
cesses are investigated. Besides, a careful investigation of its conditional distributions
also yields an exact conditional sampling algorithm, which has potential applications
of spatial extremes.
3
Association of max-stable processes to sum-stable processes
The association of α-Frechet processes to the symmetric α-stable (SαS) processes
is established in Chapter III. Namely, under mild assumptions, every α-Frechet pro-
cess can be associated to an SαS process via spectral representations. This provides
a theoretical support to the longstanding folklore that two classes of processes share
many similar structural results. However, the converse is not true. That means that
roughly speaking, the class of SαS processes has richer structures than the class of
α-Frechet processes.
The association method has become a convenient tool to translate results on
SαS processes (e.g. Rosinski [83] and Samorodnitsky [92]) to α-Frechet processes.
By the association method, many structural results on SαS processes have natural
counterparts for α-Frechet processes. See also Kabluchko [50] for an independent
treatment with different tools.
Decomposability of max-stable processes
The decomposability properties have been extensively studied for probability dis-
tributions. The notion of decomposability can be generalized to α-Frechet processes.
Namely, letting Y = Ytt∈T be an α-Frechet process as in (1.2), a natural question
is, when can we write
(1.3) Ytt∈Td=
Y (1)t ∨ Y (2)
t
t∈T
,
where Y(i) = Y (i)t t∈T , i = 1, 2, are two independent α-Frechet processes? If such
processes Y(1),Y(2) exist, what kind of α-Frechet processes can they be? To what
extent are their structures determined by Y?
A characterization of all possible α-Frechet components Y(i) is established in
Chapter IV. Furthermore, when Y is stationary, a necessary and sufficient condition
4
for its α-Frechet component to be stationary is established. This time, Y may
have but only trivial stationary α-Frechet components (scaled copies cY with c ∈
(0, 1)), and such a process is said to be indecomposable. These indecomposable
processes can be viewed as the elementary building blocks for all stationary α-Frechet
processes. Therefore, to study stationary α-Frechet processes, it suffices to focus on
the indecomposable ones. The decomposability of stationary α-Frechet processes
also provides a different point of view on the classification problem for stationary
α-Frechet processes.
Similar decomposability results also hold for sum-stable processes. This is clear
from the association point of view. In fact, we first establish the decomposability
result for sum-stable processes, and then obtain results for max-stable processes by
the association method.
Conditional sampling for max-stable random fields
Given an α-Frechet random field Ytt∈Zd , what is the conditional distribution
First, we show the ‘if’ part. Define G(x, u) := G(x, u) (given by (3.16)) on N c
and G(x, u) := f(x)1Ax(u) + c(x) on N (if Ax and c(x) are not defined, then set
G(x, u) = 0). Set Gt(x, u) = G(x, u + t) − G(x, u). Note that Gt(x, u) is another
spectral representation of Xtt∈R and for all (x, u), 1Ax(u+ t)− 1Ax
(u)t∈R
can
take at most 2 values, one of which is 0. This observation implies (3.17) with Gt(x, u)
replaced by Gt(x, u), whence Xtt∈R is max-associable.
Next, we prove the ‘only if’ part. We show that (3.17) is violated, if G(x, u) takes
more than 2 different values on (x×R)∩N c for some x ∈ X. Suppose there exist
x ∈ E, ui ∈ R such that (x, ui) ∈ N c and gxi := G(x, ui) are mutually different, for
i = 1, 2, 3. Indeed, without loss of generality we may suppose that gx1 < gx2 < gx3.
Then, by the continuity of G, there exists > 0 such that Bi := B(x, )× (ui−, ui+
) , i = 1, 2, 3 are disjoint sets with B(x, ) := y ∈ E : ρ(x, y) < , ρ is the metric
on E and
(3.18) supB1∩N
c
G(x, u) < infB2∩N
c
G(x, u) ≤ supB2∩N
c
G(x, u) < infB3∩N
c
G(x, u) .
Put t1 = u1−u2 and t2 = u3−u2. Inequality (3.18) implies thatGt1(x, u)Gt2(x, u) < 0
on B2 ∩N c. This, in view of Theorem III.13, contradicts the max-associability. We
have thus shown (3.16).
We give two classes of SαS processes, which cannot be associated to any α-Frechet
processes, according to Proposition III.16.
Example III.17 (Non-associability of linear fractional stable motions). The linear
fractional stable motions (see Ch. 7.4 in [93]) have the following spectral representa-
tions:
Xtt∈Rd=
R
a(t+ u)H−1/α
+ − uH−1/α+
+ b
(t+ u)H−1/α
− − uH−1/α−
Mα,+(du)
t∈R
.
29
HereH ∈ (0, 1), α ∈ (0, 2), H = 1/α, a, b ∈ R and |a|+|b| > 0. By Proposition III.16,
these processes are not max-associable.
Example III.18 (Non-associability of Telecom processes). The Telecom process
offers an extension of fractional Brownian motion consistent with heavy-tailed fluc-
tuations. It is a large scale limit of renewal reward processes and it can be obtained
by choosing the distribution of the rewards accordingly (see [57] and [72]). A Telecom
process Xtt∈R has the following representation
Xtt∈Rd=
R
R
e(H−1)/α (F (es(t+ u))− F (esu))Mα,+(ds, du)
t∈R
,
where 1 < α < 2, 1/α < H < 1, F (z) = (z ∧ 0 + 1)+ , z ∈ R and the SαS random
measure Mα,+ is with control measure mα(ds, du) = dsdu. By Proposition III.16,
the Telecom process is not max-associable.
Remark III.19. It is important that the index T in Proposition III.16 is the entire real
line R. Indeed, in both Example III.17 and III.18, when the time index is restricted
to the half-line T = R+ (or T = R−), the processes Xtt∈T satisfy condition (3.13)
and are therefore max-associable.
3.4 Association of Classifications
In this section, we show how to apply the association technique to relate various
classification results for SαS and α-Frechet processes. Note that, many classifications
of SαS (α-Frechet as well) processes are induced by suitable decompositions of the
measure space (S, µ). The following theorem provides an essential tool for translating
classification results for SαS to α-Frechet processes, and vice versa.
Theorem III.20. Suppose an SαS process Xtt∈T and an α-Frechet process Ytt∈T
are associated by two spectral representations f (i)t t∈T ⊂ Lα
+(Si, µi) for i = 1, 2. That
30
is,
Xtt∈Td=
Si
f (i)t dM (i)
α,+
t∈T
and Ytt∈Td=
e
Si
f (i)t dM (i)
α,∨
t∈T
, i = 1, 2 .
Then, for any measurable subsets Ai ⊂ Si, i = 1, 2, we have
A1
f (1)t dM (1)
α,+
t∈T
d=
A2
f (2)t dM (2)
α,+
t∈T
if and only if
e
A1
f (1)t dM (1)
α,∨
t∈T
d=
e
A2
f (2)t dM (2)
α,∨
t∈T
.
The proof follows from Theorem III.2 by restricting the measures onto the sets Ai, i =
1, 2.
For an SαS process Xtt∈T with spectral functions ftt∈T ⊂ Lα(S, µ), a de-
composition typically takes the form Xtt∈Td=
n
j=1 X(j)t t∈T , where X(j)
t =
A(j) ftdMα,+ for all t ∈ T and A(j), 1 ≤ j ≤ n are disjoint subsets of S =
n
j=1 A(j).
The components X(j)t t∈T , 1 ≤ j ≤ n are independent SαS processes. When
Xtt∈T is max-associable, Theorem III.20 enables us to define the associated de-
composition, for the α-Frechet process Ytt∈T associated with Xtt∈T . Namely,
we have Ytt∈Td=
n
j=1 Y(j)t t∈T , where Y (j)
t =e
A(j) |ft|dMα,∨ for all t ∈ T . Con-
versely, given a decomposition for α-Frechet processes, we can define a corresponding
decomposition for the associated SαS processes.
Example III.21 (Conservative-dissipative decomposition). In the seminal work, [83]
established the conservative-dissipative decomposition for SαS processes. Namely, for
any Xtt∈T with representation (3.1), one have
Xtt∈Td= XC
t+XD
tt∈T ,
where XC
t=
CftdMα,+ and XD
t=
DftdMα,+ for all t ∈ T , with C and D defined
31
by
(3.19) C :=s :
T
ft(s)αλ(dt) = ∞
and D := S \ C .
When Xtt∈T is stationary, the sets C and D correspond to the Hopf decom-
position S = C ∪ D of the non-singular flow associated with Xtt∈T (see [83]
for details). Therefore, XC
tt∈T and XD
tt∈T are referred to as the conservative
and dissipative components of Xtt∈T , respectively. Theorem III.20 enables us to
use (3.19) to establish the parallel decomposition of the associated α-Frechet process
Ytt∈T . Namely, for the associated Ytt∈T , we have, Ytt∈Td= Y C
t∨ Y D
tt∈T ,
where Y C
t=
e
C|ft|dMα,∨ and Y D
t=
e
D|ft|dMα,∨ for all t ∈ T . This decomposition
was established in [109] by using different tools.
Remark III.22. Similar associations can be established for other decompositions,
including positive-null decomposition (see [92] and [109]), and the decompositions of
the above two types for random fields (T = Zd or R
d, see [91] and [108]). A more
specific decomposition for SαS processes with representation (3.15) was developed
in [70], and one can obtain the corresponding decomposition for the associated α-
Frechet process by Theorem III.20.
3.5 Proofs of Auxiliary Results
We first need the following lemma.
Lemma III.23. If F ⊂ Lα
+(S, µ), then
(i) ρ(F ) = ρ(span+(F )) = ρ(∨-span(F )), and
(ii) for any f (1) ∈ span+(F ) and f (2) ∈ ∨-span(F ), f (1)/f (2) ∈ ρ(F ).
32
Proof. (i) First, for any fi, gi ∈ F, ai ≥ 0, bi ≥ 0, i ∈ N, we have
i∈N
aifij∈N
bjgj≤ x
=
i∈N
aifij∈N
bjgj≤ x
=
i∈N
k∈N
j∈N
aifibjgj
< x+1
k
,
hence ρ(∨-span(F )) ⊂ ρ(span+(F )).
To show ρ(span+(F )) ⊂ ρ(∨-span(F )), we shall first prove that ρ(span+(F )) ⊂
ρ(∨-span(F )), where span+(F ) involves only finite positive linear combinations. For
all f1, f2, g1 ∈ F, a1, b1, b2 ≥ 0, we have
a1f1 + a2f2b1g1
≤ x=
qj∈Q
a1f1b1g1
≤ qj∩
a2f2b1g1
≤ x− qj
,
This shows that (a1f1 + a2f2)/b1g1 is ρ(∨-span(F )) measurable. By using the fact
that F contains only nonnegative functions and since
b1g1
a1f1 + a2f2≤ x
=
a1f1 + a2f2
b1g1≥
1
x
, x > 0,
we similarly obtain that (a1f1+a2f2)/(b1g1+b2g2) is ρ(∨-span(F )) measurable. Sim-
ilarly arguments can be used to show that (
n
i=1 aifi)/(
n
i=1 bigi) is ρ(∨-span(F ))
measurable for all ai, bi ≥ 0, fi, gi ∈ F, 1 ≤ i ≤ n.
We have thus shown that ρ(span+(F )) ⊂ ρ(∨-span(F )). If now f, g ∈ span+(F ),
then there exist two sequences fn, gn ∈ span+(F ), such that fn → f and gn → g a.e..
Thus, hn := fn/gn → h := f/g as n → ∞, a.e.. Since hn are ρ(span+(F )) measurable
for all n ∈ N, so is h. Hence ρ(span+(F )) = ρ(span+(F )) ⊂ ρ(∨-span(F )).
(ii) By the previous argument, it is enough to focus on finite linear and max-
linear combinations. Suppose f (1) =
n
i=1 aifi and f (2) =
p
j=1 bjgj for some fi, gj ∈
F, ai, bj ≥ 0, 1 ≤ i ≤ n, 1 ≤ j ≤ p. Then, for all x > 0,
n
i=1 aifip
j=1 bjgj< x
=
p
j=1
n
i=1
aifigj
< xbj∈ ρ(F ) .
It follows that f (1)/f (2) ∈ ρ(F ).
33
Proof of Proposition III.8. First we show Re,+(F∨) ⊃ Re,+(F+), where F∨ and F+
are defined in (3.11). By (3.8), it suffices to show that, for any r2 ∈ ρ(F+), f (2) ∈ F+,
there exist r1 ∈ ρ(F∨) and f (1) ∈ F∨, such that
(3.20) r1f(1) = r2f
(2) .
To obtain (3.20), we need the concept of full support. We say a function g has full
support in F (an arbitrary collection of functions defined on (S, µ)), if g ∈ F and for
all f ∈ F , µ(supp(g)\supp(f)) = 0. Here supp(f) := s ∈ S : f(s) = 0. By Lemma
3.2 in [109], there exists function f (1) ∈ F∨, which has full support in F∨. One can
show that this function has also full support in F+. Indeed, let g ∈ F+ be arbitrary.
Then, there exist gn =
kn
i=1 anigni, ani ≥ 0 and gni ∈ F ⊂ F∨ such that gnµ
−→ g as
n → ∞. Note that µ(supp(gn) \ supp(f)) = 0 for all n. Thus, for all > 0, we have
µ(|gn − g| > ) ≥ µ(|g| > \ supp(f)). Since µ(|gn − g| > ) → 0 as n → ∞, it
follows that µ(|g| > \ supp(f)) = 0 for all > 0, i.e., µ(supp(g) \ supp(f)) = 0.
We have thus shown that f has full support in F+.
Now, set r1 := r2f (2)/f (1)
, we have (3.20). (Note that f (2) = 0 , µ-a.e. on
S \ supp(f (1)). By setting 0/0 = 0, f (2)/f (1) is well defined.) Lemma III.23 (ii)
implies that f (2)/f (1) ∈ ρ(F ), whence r1 ∈ ρ(F ) = ρ(F+). We have thus shown
Re,+(F∨) ⊃ Re,+(F+). In a similar way one can show Re,+(F∨) ⊂ Re,∨(F+).
Proof of Proposition III.9. First, suppose (3.12) does not hold but (3.5) holds. Then,
without loss of generality, we can assume that there exists S(1)0 ⊂ S1 such that
f (1)1 (s) > 0, f (1)
2 (s) < 0 for all s ∈ S(1)0 and µ(S(1)
0 ) > 0. It follows from (3.5)
that there exists a linear isometry U such that, by Theorem III.5, Uf (1)i
= f (2)i
=
T (ri)U(f), with certain f and ri = f (1)i
/f , for i = 1, 2. In particular, f can be taken
with full support. Note that sign(r1) = sign(r2) on S(1)0 . It follows that f (2)
1 and f (2)2
34
have different signs on a set of positive measure (indeed, this set is the image of the
S(1)0 under the regular set isomorphism T ). This contradicts the fact that f (2)
1 and
f (2)2 are both nonnegative on S2.
On the other hand, suppose (3.12) is true. Define Uf (1)i
:= |f (1)i
|. It follows
from (3.12) that U can be extended to a positive-linear isometry from Lα(S1, µ1) to
Lα
+(S2, µ2), which implies (3.5).
CHAPTER IV
Decomposability of Sum- and Max-stable Processes
In this chapter, we investigate the general decomposability problem for both SαS
and α-Frechet processes with 0 < α < 2. We first focus on SαS processes. Then,
by the association method introduced in Chapter III, the counterpart results for
α-Frechet processes are proved with little extra effort in Section 4.3.
Let X = Xtt∈T be an SαS process. We are interested in the case when X can
be written as
(4.1) Xtt∈Td=
X(1)
t + · · ·+X(n)t
t∈T
,
where ‘d=’ stands for ‘equality in finite-dimensional distributions’, and X(k) =
X(k)t t∈T , k = 1, . . . , n are independent SαS processes. We will write X
d= X(1) +
· · ·+X(n) in short, and each X(k) will be referred to as a component of X. The sta-
bility property readily implies that (4.1) holds with X(k) d= n−1/αX ≡ n−1/αXtt∈T .
The components equal in finite-dimensional distributions to a constant multiple of X
will be referred to as trivial. We are interested in the general structure of non-trivial
SαS components of X.
Many important decompositions (4.1) of SαS processes (with non-trivial compo-
nents) are already available in the literature: see for example Cambanis et al. [12],
Rosinski [83], Rosinski and Samorodnitsky [86], Surgailis et al. [103], Pipiras and
35
36
Taqqu [70, 71], and Samorodnitsky [92], to name a few. These results were mo-
tivated by studies of various probabilistic and structural aspects of the underlying
SαS processes such as ergodicity, mixing, stationarity, self-similarity, etc. Notably,
Rosinski [83] established a fundamental connection between stationary SαS processes
and non-singular flows. He developed important tools based on minimal represen-
tations of SαS processes and inspired multiple decomposition results motivated by
connections to ergodic theory.
In this chapter, we adopt a different perspective. Our main goal is to charac-
terize all possible SαS decompositions (4.1). Our results show how the dependence
structure of an SαS process determines the structure of its components.
Consider SαS processes Xtt∈T indexed by a complete separable metric space T
with an integral representation
(4.2) Xtt∈Td=
S
ft(s)Mα(ds)
t∈T
,
with spectral functions ftt∈T ⊂ Lα(S,BS, µ). Recall that for all n ∈ N, tj ∈ T, aj ∈
R,
(4.3) E exp− i
n
j=1
ajXtj
= exp
−
S
n
j=1
ajftj
α
dµ.
Without loss of generality, we always assume that the spectral functions ftt∈T ⊂
Lα(S,BS, µ) have full support, i.e., S = suppft, t ∈ T.
We first state the main result of this chapter. To this end, we recall that the ratio
σ-algebra of a spectral representation F = ftt∈T (of Xt) is defined as
(4.4) ρ(F ) ≡ ρft, t ∈ T := σft1/ft2 , t1, t2 ∈ T.
The following result characterizes the structure of all SαS decompositions.
37
Theorem IV.1. Suppose Xtt∈T is an SαS process (0 < α < 2) with spectral
representation
Xtt∈Td=
S
ft(s)Mα(ds)
t∈T
,
with ftt∈T ⊂ Lα(S,BS, µ). Let X(k)t t∈T , k = 1, · · · , n be independent SαS pro-
cesses.
(i) The decomposition
(4.5) Xtt∈Td=
X(1)
t + · · ·+X(n)t
t∈T
holds, if and only if there exist measurable functions rk : S → [−1, 1], k =
1, · · · , n, such that
(4.6) X(k)t t∈T
d=
S
rk(s)ft(s)Mα(ds)
t∈T
, k = 1, · · · , n.
In this case, necessarily
n
k=1 |rk(s)|α = 1, µ-almost everywhere on S.
(ii) If (4.5) holds, then the rk’s in (4.6) can be chosen to be non-negative and ρ(F )-
measurable. Such rk’s are unique modulo µ.
The rest of the chapter is structured as follows. In Section 4.1, we provide some
consequences of Theorem IV.1 for general SαS processes. The stationary case is
discussed in Section 4.2. Parallel results on max-stable processes are presented in
Section 4.3. The proof of Theorem IV.1 is given in Section 4.4.
4.1 SαS Components
In this section, we provide a few examples to illustrate the consequences of our
main result Theorem IV.1. The first one is about SαS processes with independent
increments. Recall that we always assume 0 < α < 2.
38
Corollary IV.2. Let X = Xtt∈R+ be an arbitrary SαS process with independent
increments and X0 = 0. Then all SαS components of X also have independent
increments.
Proof. Write m(t) = Xtα
α, where Xtα denotes the scale coefficient of the SαS
random variable Xt. By the independence of the increments of X, it follows that
m is a non-decreasing function with m(0) = 0. First, we consider the simple case
when m(t) is right-continuous. Consider the Borel measure µ on [0,∞) determined
by µ([0, t]) := m(t). The independence of the increments of X readily implies that
X has the representation:
(4.7) Xtt∈R+
d=
∞
0
1[0,t](s)Mα(ds)
t∈R+
,
where Mα is an SαS random measure with control measure µ.
Now, for any SαS component Y (≡ X(k)) of X, we have that (4.6) holds with
ft(s) = 1[0,t](s) and some function r(s)(≡ rk(s)). This implies that the increments of
Y are also independent since, for example, for any 0 ≤ t1 < t2, the spectral functions
r(s)ft1(s) = r(s)1[0,t1](s) and r(s)ft2(s) − r(s)ft1(s) = r(s)1(t1,t2](s) have disjoint
supports.
It remains to prove the general case. The difficulty is that m(t) may have (at
most countably many) discontinuities, and a representation as (4.7) is not always
possible. Nevertheless, introduce the right-continuous functions t → mi(t), i = 0, 1,
m0(t) := m(t+)−
τ≤t
(m(τ)−m(τ−)) and m1(t) :=
τ≤t
(m(τ)−m(τ−))
and let Mα be an SαS random measure on R+×0, 1 with control measure µ([0, t]×
i) := mi(t), i = 0, 1, t ∈ R+. In this way, as in (4.7) one can show that
Xtt∈Td=
R+×0,1
1[0,t)×0(s, v) + 1[0,t]×1(s, v)Mα(ds, dv)
t∈T
.
39
The rest of the proof remains similar and is omitted.
Remark IV.3. Theorem IV.1 and Corollary IV.2 do not apply to the Gaussian case
(α = 2). For the sake of simplicity, take T = 1, 2 and n = 2 (2 SαS components)
in (4.1). In this case, all the (in)dependence information of the mean-zero Gaussian
process Xtt∈T is characterized by the covariance matrix Σ of the Gaussian vector
(X(1)1 , X(2)
1 , X(1)2 , X(2)
2 ). A counterexample can be easily constructed by choosing
appropriately Σ. This reflects the drastic difference of the geometries of Lα spaces
for α < 2 and α = 2.
The next natural question to ask is whether two SαS processes have common
components. Namely, the SαS process Z is a common component of the SαS processes
X and Y , if Xd= Z + X(1) and Y
d= Z + Y (1), where X(1) and Y (1) are both SαS
processes independent of Z.
To study the common components, the co-spectral point of view introduced in
Wang and Stoev [111] is helpful. Consider a measurable SαS process Xtt∈T with
spectral representation (4.2), where the index set T is equipped with a measure λ
defined on the σ-algebra BT . Without loss of generality, we take f(·, ·) : (S×T,BS ×
BT ) → (R,BR) to be jointly measurable (see Theorems 9.4.2 and 11.1.1 in [93]).
The co-spectral functions, f·(s) ≡ f(s, ·), are elements of L0(T ) ≡ L0(T,BT ,λ), the
space of BT -measurable functions modulo λ-null sets. The co-spectral functions are
indexed by s ∈ S, in contrast to the spectral functions ft(·) indexed by t ∈ T . Recall
also that a set P ⊂ L0(T ) is a cone, if cP = P for all c ∈ R \ 0 and 0 ∈ P . We
write f·(s)s∈S ⊂ P modulo µ, if for µ-almost all s ∈ S, f·(s) ∈ P .
Proposition IV.4. Let X(i) = X(i)t t∈T be SαS processes with measurable represen-
tations f (i)t t∈T ⊂ Lα(Si,BSi
, µi), i = 1, 2. If there exist two cones Pi ⊂ L0(T ), i =
40
1, 2, such that f (i)· (s)s∈Si
⊂ Pi modulo µi, for i = 1, 2, and P1 ∩ P2 = 0, then
the two processes have no common component.
Proof. Suppose Z is a component of X(1). Then, by Theorem IV.1, Z has a spectral
representation r(1)f (1)t t∈T , for some BS1-measurable function r(1). By the definition
of cones, the co-spectral functions of Z are included in P1, i.e., r(1)(s)f(1)· (s)s∈S1 ⊂
P1 modulo µ1. If Z is also a component of X(2), then by the same argument,
r(2)(s)f (2)· (s)s∈S2 ⊂ P2 modulo µ2, for some BS2-measurable function r(2)(s). Since
P1∩P2 = 0, it then follows that µi(supp(r(i))) = 0, i = 1, 2, or equivalently Z = 0,
the degenerate case.
We conclude this section with an application to SαS moving averages.
Corollary IV.5. Let X(1)and X(2)
be two SαS moving averages
X(i)t t∈Rd
d=
Rd
f (i)(t+ s)M (i)α(ds)
t∈Rd
with kernel functions f (i) ∈ Lα(Rd,BRd ,λ), i = 1, 2. Then, either
(4.8) X(1) d= cX(2)
for some c > 0 ,
or X(1)and X(2)
have no common component. Moreover, (4.8) holds, if and only if
for some τ ∈ Rdand ∈ ±1,
(4.9) f (1)(s) = cf (2)(s+ τ) , µ-almost all s ∈ S.
Proof. Clearly (4.9) implies (4.8). Conversely, if (4.8) holds, then (4.9) follows as in
the proof of Corollary 4.2 in [111], with slight modification (the proof therein was
for positive cones). When (4.8) (or equivalently (4.9)) does not hold, consider the
smallest cones containing f (i)(s+ ·)s∈R, i = 1, 2 respectively. Since these two cones
have trivial intersection 0, Proposition IV.4 implies that X(1) and X(2) have no
common component.
41
4.2 Stationary SαS Components and Flows
LetX = Xtt∈T be a stationary SαS process with representation (4.2), where now
T = Rd or T = Z
d, d ∈ N. The seminal work of Rosnski [83] established an important
connection between stationary SαS processes and flows. A family of functions φtt∈T
is said to be a flow on (S,BS, µ), if for all t1, t2 ∈ T , φt1+t2(s) = φt1(φt2(s)) for all
s ∈ S, and φ0(s) = s for all s ∈ S. We say that a flow is non-singular, if µ(φt(A)) = 0
is equivalent to µ(A) = 0, for all A ∈ BS, t ∈ T . Given a flow φtt∈T , ctt∈T is
said to be a cocycle if ct+τ (s) = ct(s)cτ φt(s) µ-almost surely for all t, τ ∈ T and
ct ∈ ±1 for all t ∈ T .
To understand the relation between the structure of stationary SαS processes
and flows, it is necessary to work with minimal representations of SαS processes,
introduced by Hardin [45, 46]. The minimality assumption is crucial in many results
on the structure of SαS processes, although it is in general difficult to check (see
e.g. Rosinski [85] and Pipiras [69]).
Definition IV.6. The spectral functions F ≡ ftt∈T (and the corresponding spec-
tral representation (4.2)) are said to be minimal, if the ratio σ-algebra ρ(F ) in (4.4)
is equivalent to BS, i.e., for all A ∈ BS, there exists B ∈ ρ(F ) such that µ(A∆B) = 0,
where A∆B = (A \B) ∪ (B \ A).
Rosinski ([83], Theorem 3.1) proved that if ftt∈T is minimal, then there exists
a modulo µ unique non-singular flow φtt∈T , and a corresponding cocycle ctt∈T ,
such that for all t ∈ T ,
(4.10) ft(s) = ct(s)dµ φt
dµ(s)
1/αf0 φt(s) , µ-almost everywhere.
Conversely, suppose that (4.10) holds for some non-singular flow φtt∈T , a cor-
responding cocycle ctt∈T , and a function f0 ∈ Lα(S, µ) (ftt∈T not necessarily
42
minimal). Then, clearly the SαS process X in (4.2) is stationary. In this case, we
shall say that X is generated by the flow φtt∈T .
Consider now an SαS decomposition (4.1) of X, where the independent com-
ponents X(k)t t∈T ’s are stationary. This will be referred to as a stationary SαS
decomposition, and the X(k)t t∈T ’s as stationary components of X. Our goal in this
section is to characterize the structure of all possible stationary components. This
characterization involves the invariant σ-algebra with respect to the flow φtt∈T :
(4.11) Fφ = A ∈ BS : µ(φτ (A)∆A) = 0 , for all τ ∈ T .
Given a function g and a σ-algebra G, we write g ∈ G, if g is measurable with respect
to G.
Theorem IV.7. Let Xtt∈T be a stationary and measurable SαS process with spec-
tral functions ftt∈T given by
ft(s) =
S
ct(s)dµ φt
dµ(s)
1/αf0 φt(s)Mα(ds), t ∈ T .
(i) Suppose that Xtt∈T has a stationary SαS decomposition
(4.12) Xtt∈Td=
X(1)
t + · · ·+X(n)t
t∈T
.
Then, each component X(k)t t∈T has a representation
(4.13) X(k)t t∈T
d=
S
rk(s)ft(s)Mα(ds)
t∈T
, k = 1, · · · , n,
where the rk’s can be chosen to be non-negative and ρ(F )-measurable. This choice is
unique modulo µ and these rk’s are φ-invariant, i.e. rk ∈ Fφ.
(ii) Conversely, for any φ-invariant rk’s such that
n
k=1 |rk(s)|α = 1, µ-almost ev-
erywhere on S, decomposition (4.12) holds with X(k)’s as in (4.13).
43
Proof. By using (4.10), a change of variables, and the φ-invariance of the functions
rk’s, one can show that the X(k)’s in (4.13) are stationary. This fact and Theorem
IV.1 yield part (ii).
We now show (i). Suppose that X(k) is a stationary (SαS) component of X.
Theorem IV.1 implies that there exists unique modulo µ non-negative and ρ(F )-
measurable function rk for which (4.13) holds. By the stationarity of X(k), it also
follows that for all τ ∈ T , rk(s)ft+τ (s)t∈T is also a spectral representation of X(k).
By the flow representation (4.10), it follows that for all t, τ ∈ T ,
(4.14) ft+τ (s) = cτ (s)ft φτ (s)dµ φτ
dµ
1/α(s) , µ-almost everywhere,
and we obtain that for all τ, tj ∈ T, aj ∈ R, j = 1, · · · , n:
S
n
j=1
ajrk(s)ftj+τ (s)α
µ(ds) =
S
n
j=1
ajrk φ−τ (s)ftj(s)α
µ(ds),
which shows that rk φ−τ (s)ft(s)t∈T is also a representation for X(k), for all τ ∈ T .
Observe that from (4.14), for all t1, t2, τ ∈ T and λ ∈ R,
ft1+τ
ft2+τ
≤ λ= φ−1
τ
ft1ft2
≤ λmodulo µ.
It then follows that for all τ ∈ T , the σ-algebra φ−τ (ρ(F )) ≡ (φτ )−1(ρ(F )) is equiv-
alent to ρ(F ). This, by the uniqueness of rk ∈ ρ(F ) (Theorem IV.1), implies that
rk φτ = rk modulo µ, for all τ . Then, rk ∈ Fφ follows from standard measure-
theoretic argument. The proof is complete.
Remark IV.8. The structure of the stationary SαS components of stationary SαS pro-
cesses (including random fields) has attracted much interest since the seminal work
of Rosinski [83, 84]. See, for example, Pipiras and Taqqu [71], Samorodnitsky [92],
Roy [87, 88], Roy and Samorodnitsky [91], Roy [89, 90], and Wang et al. [108]. In
44
view of Theorem IV.7, the components considered in these works correspond to in-
dicator functions rk(s) = 1Ak(s) of certain disjoint flow-invariant sets Ak’s arising
from ergodic theory (see e.g. Krengel [55] and Aaronson [1]).
Theorem IV.7 can be applied to check indecomposability of stationary SαS pro-
cesses. Recall that a stationary SαS process is said to be indecomposable, if all its
stationary SαS components are trivial (i.e. constant multiples of the original process).
Corollary IV.9. Consider Xtt∈T as in Theorem IV.7. If Fφ is trivial, then
Xtt∈T is indecomposable. The converse is true when, in addition, ftt∈T is mini-
mal.
Proof. If Fφ is trivial, the result follows from Theorem IV.7. Conversely, let ftt∈T
be minimal and X indecomposable. Then, one can choose A ∈ Fφ, such that µ(A) >
0 and µ(S \ A) > 0. Then, consider
XA
tt∈T
d=
S
1A(s)ft(s)Mα(ds)
t∈T
.
By Theorem IV.7, XA is a stationary component of X. It suffices to show that XA
is a non-trivial of X, which would contradict the indecomposability.
Suppose that XA is trivial, then cXA d= X, for some c > 0. Thus, by Theorem
IV.7, cXA has a representation as in (4.13), with rk := c1A. On the other hand, since
cXA d= X, we also have the trivial representation with rk := 1. Since A ∈ ρ(F ),
the uniqueness of rk implies that 1 = c1A modulo µ, which contradicts µ(Ac) > 0.
Therefore, XA is non-trivial.
The indecomposable stationary SαS processes can be seen as the elementary build-
ing blocks for the construction of general stationary SαS processes. We conclude this
section with two examples.
45
Example IV.10 (Mixed moving averages). Consider a mixed moving average in the
sense of [102]:
(4.15) Xtt∈Rd
d=
Rd×V
f(t+ s, v)Mα(ds, dv)
t∈Rd
.
Here, Mα is an SαS random measure on Rd×V with the control measure λ×ν, where
λ is the Lebesgue measure on (Rd,BRd) and ν is a probability measure on (V,BV ), and
f(s, v) ∈ Lα(Rd × V,BRd×V ,λ× ν). Given a disjoint union V =
n
j=1 Aj, where Aj’s
are measurable subsets of V , the mixed moving averages can clearly be decomposed
as in (4.12) with
X(k)t t∈Rd
d=
Rd×Ak
f(t+ s, v)Mα(ds, dv)
t∈Rd
, for all k = 1, . . . , n .
Any moving average process
(4.16) Xtt∈Rd
d=
Rd
f(t+ s)Mα(ds)
t∈Rd
trivially has a mixed moving average representation. The next result shows when
the converse is true.
Corollary IV.11. The mixed moving average X in (4.15) is indecomposable, if and
only if it has a moving average representation as in (4.16).
Proof. By Corollary IV.9, the moving average process (4.16) is indecomposable, since
in this case φt(s) = t + s, t, s ∈ Rd and therefore Fφ is trivial. This proves the ‘if’
part.
Suppose now thatX in (4.15) is indecomposable. In Section 5 of Pipiras [69] it was
shown that SαS processes with mixed moving average representations and stationary
increments also have minimal representations of the mixed moving average type. By
using similar arguments, one can show that this is also true for the class of stationary
mixed moving average processes.
46
Thus, without loss of generality, we assume that the representation in (4.15) is
minimal. Suppose now that there exists a set A ∈ BV with ν(A) > 0 and ν(Ac) > 0.
Since Rd × A and R
d × Ac are flow-invariant, we have the stationary decomposition
Xtt∈Rd
d= XA
t+XA
c
tt∈Rd , where
XB
t:=
R×V
1B(v)f(t+ s, v)Mα(ds, dv), B ∈ A,Ac.
Note that both components XA = XA
tt∈Rd and XA
c
= XAc
tt∈Rd are non-zero
because the representation of X has full support.
Now, since X is indecomposable, there exist positive constants c1 and c2, such
that Xd= c1XA d
= c2XAc
. The minimality of the representation and Theorem IV.7
imply that c11A = c21Ac modulo ν, which is impossible. This contradiction shows
that the set V cannot be partitioned into two disjoint sets of positive measure. That
is, V is a singleton and the mixed moving average is in fact a moving average.
Example IV.12 (Doubly stationary processes). Consider a stationary process ξ =
ξtt∈T (T = Zd) supported on the probability space (E, E , µ) with ξt ∈ Lα(E, E , µ).
Without loss of generality, we may suppose that ξt(u) = ξ0 φt(u), where φtt∈T is
a µ-measure-preserving flow.
Let Mα be an SαS random measure on (E, E , µ) with control measure µ. The
stationary SαS process X = Xtt∈T
(4.17) Xt :=
E
ξt(u)Mα(du), t ∈ T
is said to be doubly stationary (see Cambanis et al. [11]). By Corollary IV.9, if ξ is
ergodic, then X is indecomposable.
A natural and interesting question raised by a referee is: what happens when X
is decomposable and hence ξ is non-ergodic? Can we have a direct integral decom-
47
position of the process X into indecomposable components? The following remark
partly addresses this question.
Remark IV.13. The doubly stationary SαS processes are a special case of station-
ary SαS processes generated by positively recurrent flows (actions). As shown in
Samorodnitsky [92], Remark 2.6, each such stationary SαS process X = Xtt∈T can
be expressed through a measure-preserving flow (action) on a finite measure space.
Namely,
(4.18) Xtt∈Td=
E
ft(u)M(µ)α
(du)
t∈T
, with ft(u) := ct(u)f0 φt(u),
where M (µ)α is an SαS random measure with a finite control measure µ on (E, E),
φ = φtt∈T is a µ-preserving flow (action), and ctt∈T is a co-cycle with respect to
φ. In the case when the co-cycle is trivial (ct ≡ 1) and µ(E) = 1, the process X is
doubly stationary.
For simplicity, suppose that T = Zd and without loss of generality let (E, E , µ)
be a standard Lebesgue space with µ(E) = 1. The ergodic decomposition theorem
(see e.g. Keller [52], Theorem 2.3.3) implies that there exists conditional probability
distributions µuu∈E with respect to I such that φ is measure-preserving and ergodic
with respect to the measures µu for µ-almost all u ∈ E. Let ν be another φ-invariant
measure on (E, E) dominating the conditional probabilities µu so that the Radon–
Nikodym derivatives p(x, u) = (dµu/dν)(x) are jointly measurable on (E × E, E ⊗
E , ν × µ). Consider
gt(x, u) = ft(x)p(φt(x), u)1/α.
Recall that ν and µu are φ-invariant, whence
p(φt(x), u) =dµu
dν(φt(x)) =
dµu
dν(x) = p(x, u), modulo ν × µ.
48
Thus, gt(x, u) = ft(x)(dµu/dν)1/α(x), and for all aj ∈ R, tj ∈ T, j = 1, · · · , n, we
have
E2
n
j=1
ajgtj(x, u)α
ν(dx)µ(du) =
E2
n
j=1
ajftj(x)αdµu
dν(x)ν(dx)µ(du)
=
E2
n
j=1
ajftj(x)α
dµu(dx)µ(du)
=
E
n
j=1
ajftj(x)α
µ(dx),
where the last equality follows from the identity that
E
h(x)µ(dx) =
E2
h(x)µu(dx)µ(du), for all h ∈ L1(E, E , µ).
We have thus shown that Xtt∈T defined by (4.18) has another spectral represen-
tation
(4.19) Xtt∈Td=
E×E
gt(x, u)M(ν×µ)α
(dx, du)
t∈T
,
where M (ν×µ)α is an SαS random measure on E × E with control measure ν × µ. It
also follows that for µ-almost all u ∈ E, the process defined by
X(u)t
:=
E
gt(x, u)M(ν)α
(dx), t ∈ T,
is indecomposable, where M (ν)α has control measure ν. Indeed, as above, one can
show that
X(u)t t∈T
d=
E
ft(u, x)M(µu)α
(dx)
t∈T
,
where M (µu)α has control measure µu. The ergodic decomposition theorem implies
that the flow (action) φ is ergodic with respect to µu, which by Corollary IV.9 implies
the indecomposability of X(u) = X(u)t t∈T . In this way, (4.19) parallels the mixed
moving average representation for stationary SαS processes generated by dissipative
flows (see e.g. Rosinski [83]).
49
Remark IV.14. The above construction of the decomposition (4.19) assumes the
existence of a φ-invariant measure ν dominating all conditional probabilities µu, u ∈
E. If the measure µ, restricted on the invariant σ-algebra Fφ is discrete, i.e. Fφ
consists of countably many atoms under µ, then one can take ν ≡ µ. In this case,
the process X is decomposed into a sum (possibly infinite) of its indecomposable
components:
Xt =
k
Ek
ft(x)M(µ)α
(dx),
where the Ek’s are disjoint φ-invariant measurable sets, such that E = ∪kEk and
φ|Ekis ergodic, for each k. In this case, the Ek’s are the atoms of Fφ.
In general, when µ|Fφis not discrete, the dominating measure ν if it exists, may not
be σ-finite. Indeed, since the φt’s are ergodic for µu, it follows that either µu = µu
or µu and µu are singular, for µ-almost all u, u ∈ E. Thus, if Fφ is “too rich”,
this singularity feature implies that the measure ν may not be chosen to be σ-finite.
4.3 Decomposability of Max-stable Processes
In this section, we state and prove some results on the (max-)decomposability of
max-stable processes. Again, we focus on α-Frechet processes.
Let Y = Ytt∈T be an α-Frechet process. If
(4.20) Ytt∈Td=
Y (1)t ∨ · · · ∨ Y (n)
t
t∈T
,
for some independent α-Frechet processes Y (k) = Y (k)t t∈T , i = 1, · · · , n, then we
say that the Y (k)’s are components of Y . By the max-stability of Y , (4.20) trivially
holds if the Y (k)’s are independent copies of n−1/αYtt∈T . The constant multiples of
Y are referred to as trivial components of Y and as in the SαS case, we are interested
in the structure of the non-trivial ones.
50
The association method can be readily applied to transfer decomposability results
for SαS processes to the max-stable setting. Let Y = Ytt∈T be an α-Frechet
(α ∈ (0, 2)) process with extremal representation
(4.21) Ytt∈Td=
e
S
ft(s)M∨
α(ds)
t∈T
,
where ftt∈T ⊂ Lα
+(S,BS, µ) are spectral functions, and recall that
(4.22) P(Yti≤ yi, i = 1, · · · , n) = exp
−
S
max1≤i≤n
fti(s)
yi
α
µ(ds),
for all yi > 0, ti ∈ T, i = 1, · · · , n.
Assume 0 < α < 2. Recall that, an SαS process X and an α-Frechet process Y
are said to be associated if they have a common spectral representation. That is, if
for some non-negative ftt∈T ⊂ Lα
+(S,BS, µ), Relations (4.2) and (4.21) hold.
To illustrate the association method in Chapter III, we prove the max-stable
counterpart of our main result Theorem IV.1. From the proof, we can see that the
other results in the sum-stable setting have their natural max-stable counterparts by
association. We briefly state some of these results at the end of this section.
Theorem IV.15. Suppose Ytt∈T is an α-Frechet process with spectral represen-
tation (4.21), where F ≡ ftt∈T ⊂ Lα
+(S,BS, µ). Let Y (k)t t∈T , k = 1, · · · , n, be
independent α-Frechet processes. Then the decomposition (4.20) holds, if and only
if there exist measurable functions rk : S → [0, 1], k = 1, · · · , n, such that
(4.23) Y (k)t t∈T
d=
e
S
rk(s)ft(s)M∨
α(ds)
t∈T
, k = 1, · · · , n.
In this case,
n
k=1 rk(s)α = 1, µ-almost everywhere on S and the rk’s in (4.23) can
be chosen to be ρ(F )-measurable, uniquely modulo µ.
Proof. The ‘if’ part follows from straight-forward calculation of the cumulative distri-
bution functions (4.22). To show the ‘only if’ part, suppose (4.20) holds and Y (k) has
51
spectral functions g(k)t t∈T ⊂ Lα
+(Vk,BBk, νk), k = 1, . . . , n. Without loss of general-
ity, assume Vkk=1,...,n to be mutually disjoint and define gt(v) =
n
k=1 g(k)t (v)1Vk
∈
Lα
+(V,BV , ν) for appropriately defined (V,BV , ν) (see the proof of Theorem IV.1).
Now, consider the SαS processX associated to Y . It has spectral functions ftt∈T
and gtt∈T . Consider the SαS processesX(k) associated to Y (k) via spectral functions
g(k)t t∈T for k = 1, . . . , n. By checking the characteristic functions, one can show
that X(k)k=1,...,n form a decomposition of X as in (4.1). Then, by Theorem IV.1,
each SαS component X(k) has a spectral representation (4.6) with spectral functions
rkftt∈T . But we introduced X(k) as the SαS process associated to Y (k) via spectral
representation g(k)t t∈T . Hence, X(k) has spectral functions g(k)t t∈T and rkftt∈T ,
and so does Y (k) by Theorem III.11. Therefore, (4.23) holds and the rest of the
desired results follow.
Further parallel results can be established by the association method. Consider
a stationary α-Frechet process Y . If Y (k), k = 1, . . . , n are independent stationary
α-Frechet processes such that (4.20) holds, then we say each Y (k) is a stationary α-
Frechet component of Y . The process Y is said to be indecomposable, if it has no non-
trivial stationary component. The following results on (mixed) moving maxima (see
e.g. [101] and [50] for more details) follow from Theorem IV.15 and the association
method, in parallel to Corollary IV.11 on (mixed) moving averages in the sum-stable
setting.
Corollary IV.16. The mixed moving maxima process
Ytt∈Rd
d=
e
Rd×V
f(t+ s, v)M∨
α(ds, dv)
t∈Rd
is indecomposable, if and only if it has a moving maxima representation
Ytt∈Rd
d=
e
Rd
f(t+ s)M∨
α(ds)
t∈Rd
.
52
4.4 Proof of Theorem IV.1
We will first show that Theorem IV.1 is true when ftt∈T is minimal (Proposi-
tion IV.18), and then we complete the proof by relating a general spectral representa-
tions to a minimal one. This technique is standard in the literature of representations
of SαS processes (see e.g. Rosinski [83], Remark 2.3). We start with a useful lemma.
Lemma IV.17. Let ftt∈T ⊂ Lα(S,BS, µ) be a minimal representation of an SαS
process. For any two bounded BS-measurable functions r(1) and r(2), we have
S
r(1)ftdMα
t∈T
d=
S
r(2)ftdMα
t∈T
,
if and only if |r(1)| = |r(2)| modulo µ.
Proof. The ’if’ part is trivial. We shall prove now the ’only if’ part. Let S(k) :=
supp(r(k)), k = 1, 2 and note that since ftt∈T is minimal, then r(k)ftt∈T , are
minimal representations, restricted to S(k), k = 1, 2, respectively. Since the latter
two representations correspond to the same process, by Theorem 2.2 in [83], there
exist a bi-measurable, one-to-one and onto point mapping Ψ : S(1) → S(2) and a
function h : S(1) → R \ 0, such that, for all t ∈ T ,
(4.24) r(1)(s)ft(s) = r(2) Ψ(s)ft Ψ(s)h(s) , almost all s ∈ S(1),
and
(4.25)dµ Ψ
dµ= |h|α , µ-almost everywhere.
It then follows that, for almost all s ∈ S(1),
(4.26)ft1(s)
ft2(s)=
r(1)(s)ft1(s)
r(1)(s)ft2(s)=
ft1 Ψ(s)
ft2 Ψ(s).
53
Define Rλ(t1, t2) = s : ft1(s)/ft2(s) ≤ λ and note that by (4.26), for all A ≡
Rλ(t1, t2),
(4.27) µ(Ψ(A ∩ S(1))∆(A ∩ S(2))) = 0 .
In fact, one can show that Relation (4.27) is also valid for all A ∈ ρ(F ) ≡
σ(Rλ(t1, t2) : λ ∈ R, t1, t2 ∈ T ). Then, by minimality, (4.27) holds for all
A ∈ BS. In particular, taking A equal to S(1) and S(2), respectively, it follows
that µ(S(1)∆S(2)) = 0. Therefore, writing S := S(1) ∩ S(2), we have
(4.28) µ(Ψ(A ∩ S)∆(A ∩ S)) = 0, for all A ∈ BS .
This implies that Ψ(s) = s, for µ-almost all s ∈ S. To see this, let BS = BS ∩ S
denote the σ-algebra BS restricted to S. Observe that for all A ∈ BS, we have
1A = 1A Ψ, for µ-almost all s ∈ S, and trivially σ(1A : A ∈ BS) = BS. Thus,
by the second part of Proposition 5.1 in [85], it follows that Ψ(s) = s modulo µ on
S. This and (4.25) imply that h(s) ∈ ±1, almost everywhere. Plugging Ψ and h
into (4.24) yields the desired result.
Proposition IV.18. Theorem IV.1 is true when ftt∈T is minimal.
Proof. We first prove the ’if ’ part. The result follows readily by using characteristic
functions. Indeed, suppose that the X(k) = X(k)t t∈T , k = 1, . . . , n are independent
and have representations as in (4.6). Then, for all aj ∈ R, tj ∈ T, j = 1, · · · ,m, we
have
(4.29) E expi
m
j=1
ajXtj
= exp
−
S
m
j=1
ajftj
α
dµ
=n
k=1
exp−
S
m
j=1
ajrkftj
α
dµ=
n
k=1
E expi
m
j=1
ajX(k)tj
,
54
where the second equality follows from the fact that
n
k=1 |rk(s)|α = 1, for µ-almost
all s ∈ S. Relation (4.29) implies the decomposition (4.1).
We now prove the ’only if ’ part. Suppose that (4.1) holds and let f (k)t t∈T ⊂
Lα(Vk,BVk, νk), k = 1, . . . , n be representations for the independent components
X(k)t t∈T , k = 1, . . . , n, respectively, and without loss of generality, assume that
Vkk=1,...,n are mutually disjoint. Introduce the measure space (V,BV , ν), where
V :=
n
k=1 Vk, BV :=
n
k=1 Ak, Ak ∈ BVk, k = 1, . . . , n and ν(A) :=
n
k=1 νk(A ∩
Vk) for all A ∈ BV .
By decomposition (4.1), it follows that Xtt∈Td=
VgtdMαt∈T , with gt(u) :=
n
k=1 f(k)t (u)1Vk
(u) andMα an SαS random measure on (V,BV ) with control measure
ν.
Thus, ftt∈T ⊂ Lα(S,BS, µ) and gtt∈T ⊂ Lα(V,BV , ν) are two representations
of the same process X, and by assumption the former is minimal. Therefore, by
Remark 2.5 in [83], there exist modulo ν unique functions Φ : V → S and h : V →
R \ 0, such that, for all t ∈ T ,
(4.30) gt(u) = h(u)ft Φ(u) , almost all u ∈ V ,
where moreover µ = νh Φ−1 with dνh = |h|αdν.
Recall that V is the union of mutually disjoint sets Vkk=1,...,n. For each k =
1, . . . , n, let Φk : Vk → Sk := Φ(Vk) be the restriction of Φ to Vk, and define the
measure µk(·) := νh,k Φ−1k( · ∩ Sk) on (S,BS) with dνh,k := |h|αdνk. Note that
µk has support Sk, and the Radon–Nikodym derivative dµk/dµ exists. We claim
that (4.6) holds with rk := (dµk/dµ)1/α. To see this, observe that for all m ∈
N, a1, . . . , am ∈ R, t1, . . . , tm ∈ T ,
S
m
j=1
ajrkftj
α
dµ =
Sk
m
j=1
ajftj
α
dµk =
Vk
m
j=1
ajhftj Φk
α
dνk ,
55
which, combined with (4.30), yields (4.6) because gt|Vk= f (k)
t .
Note also that
n
k=1 µk = µ and thus
n
k=1 rα
k= 1. This completes the proof of
part (i) of Theorem IV.1 in the case when ftt∈T is minimal.
To prove part (ii), note that the rk’s above are in fact non-negative and BS-
measurable. Note also that by minimality, the rk’s have versions rk’s that are ρ(F )-
measurable, i.e. rk = rk modulo µ. Their uniqueness follows from Lemma IV.17.
Proof of Theorem IV.1. (i) The ‘if’ part follows by using characteristic functions as
in the proof of Proposition IV.18 above.
Now, we prove the ‘only if’ part. Let ftt∈T ⊂ Lα(S,BS, µ) be a minimal represen-
tation of X. As in the proof of Proposition IV.18, by Remark 2.5 in [83], there exist
modulo µ unique functions Φ : S → S and h : S → R \ 0, such that, for all t ∈ T ,
(4.31) ft(s) = h(s) ft Φ(s) , almost all s ∈ S,
where µ = µh Φ−1 with dµh = |h|αdµ.
Now, by Proposition IV.18, if the decomposition (4.1) holds, then there exist
unique non-negative functions rk, k = 1, · · · , n, such that
(4.32) X(k)t t∈T
d=
Srk ftdMα
t∈T
, k = 1, · · · , n,
and
n
k=1 rαk = 1 modulo µ. Here Mα is an SαS measure on (S,BS) with control
measure µ. Let rk(s) := rk Φ(s) and note that by using (4.31) and a change of
variables, for all aj ∈ R, tj ∈ T, j = 1, · · · ,m, we obtain
(4.33)
S
m
j=1
ajrk(s)ftj(s)µ(ds) =
S
m
j=1
ajrk(s) ftj(s)µ(ds).
This, in view of Relation (4.32), implies (4.6). Further, the fact that
n
k=1 rαk = 1
implies
n
k=1 rα
k= 1, modulo µ, because the mapping Φ is non-singular, i.e. µΦ−1 ∼
µ. This completes the proof of part (i).
56
We now focus on proving part (ii). Suppose that (4.6) holds for two choices of rk,
namely rkand r
k. Let also r
kand r
kbe non-negative and measurable with respect
to ρ(F ). We claim that
(4.34) ρ(F ) ∼ Φ−1(ρ( F ))
and defer the proof to the end. Then, since the minimality implies that BS ∼ ρ( F ).
rkand r
kare measurable with respect to ρ(F ) ∼ Φ−1(BS). Now, Doob–Dynkin’s
lemma (see e.g. Rao [75], p. 30) implies that
(4.35) rk(s) = r
k Φ(s) and r
k(s) = r
k Φ(s), for µ almost all s,
where rkand r
kare two BS-measurable functions. By using the last relation and a
change of variables, we obtain that (4.33) holds with (rk, rk) replaced by (rk, r
k) and
(rk, r
k), respectively. Thus both r
kftt∈T and r
kftt∈T are representations of the
k-th component of X. Since ftt∈T is a minimal representation of X, Lemma IV.17
implies that rk= r
kmodulo µ. This, by (4.35) and the non-singularity of Φ yields
rk= r
kmodulo µ.
It remains to prove (4.34) Relation (4.31) and the fact that h(s) = 0 imply that
for all λ and t1, t2 ∈ T , ft1/ft2 ≤ λ = Φ−1( ft1/ ft2 ≤ λ) modulo µ. Thus the
classes of sets C := ft1/ft2 ≤ λ, t1, t2 ∈ T, λ ∈ R and C := Φ−1( ft1/ ft2 ≤
λ), t1, t2 ∈ T, λ ∈ R are equivalent. That is, for all A ∈ C, there exists A ∈ C,
with µ(A∆ A) = 0 and vice versa.
Define
G =Φ−1(A) : A ∈ ρ( F ) such thatµ(Φ−1(A)∆B) = 0 for some B ∈ σ(C)
.
Notice that G is a σ-algebra and since C ⊂ G ⊂ Φ−1(ρ( F )), we obtain that σ(C) =
Φ−1(ρ( F )) ≡ G. This, in view of definition of G, shows that for all A ∈ σ(C), exists
57
A ∈ σ(C) with µ(A∆ A) = 0. In a similar way one can show that each element of
σ(C) is equivalent to an element in σ(C), which completes the proof of the desired
equivalence of the σ-algebras.
CHAPTER V
Conditional Sampling for Max-stable Processes
The modeling and parameter estimation of the univariate marginal distributions
of the extremes have been studied extensively (see e.g. Davison and Smith [21], de
Haan and Ferreira [24], Resnick [77] and the references therein). Many of the recent
developments of statistical inference in extreme value theory focus on the character-
ization, modeling and estimation of the dependence for multivariate extremes. In
this context, building adequate max-stable processes and random fields plays a key
role. See for example de Haan and Pereira [25], Buishand et al. [9], Schlather [94],
Schlather and Tawn [95], Cooley et al. [17], and Naveau et al. [64].
This chapter is motivated by an important and long-standing challenge, namely,
the prediction for max-stable random processes and fields. Suppose that one already
has a suitable max-stable model for the dependence structure of a random field
Xtt∈T . The field is observed at several locations t1, . . . , tn ∈ T and one wants to
predict the values of the field Xs1 , . . . , Xsmat some other locations. The optimal
predictors involve the conditional distribution of Xtt∈T , given the data. Even
if the finite-dimensional distributions of the field Xtt∈T are available in analytic
form, it is typically impossible to obtain a closed-form solution for the conditional
distribution. Naıve Monte Carlo approximations are not practical either, since they
58
59
involve conditioning on events of infinitesimal probability, which leads to mounting
errors and computational costs.
Prior studies of Davis and Resnick [19, 20] and Cooley et al. [17], among others,
have shown that the prediction problem in the max-stable context is challenging,
and it does not have an elegant analytical solution. On the other hand, the growing
popularity and the use of max-stable processes in various applications, make this an
important problem. This motivated us to seek a computational solution.
5.1 Overview
In this chapter, we develop theory and methodology for sampling from the con-
ditional distributions of spectrally discrete max-stable models. More precisely, we
provide an algorithm that can generate efficiently exact independent samples from the
regular conditional probability of (Xs1 , . . . , Xsm), given the values (Xt1 , . . . , Xtn
). For
the sake of simplicity, we write X = (X1, . . . , Xn) ≡ (Xt1 , . . . , Xtn). The algorithm
applies to the general max-linear model:
(5.1) Xi = maxj=1,...,p
ai,jZj ≡
p
j=1
ai,jZj , i = 1, . . . , n.
where the ai,j’s are known non-negative constants and the Zj’s are independent
continuous non-negative random variables. Any multivariate max-stable distribution
can be approximated arbitrarily well via a max-linear model with sufficiently large
p (see e.g. Remark II.1).
The main idea is to first generate samples from the regular conditional probability
distribution of Z | X = x, where Z = (Zj)j=1,...,p. Then, the conditional distributions
of
Xsk=
p
j=1
bk,jZj , k = 1, . . . ,m,
60
given X = x can be readily obtained, for any given bk,j’s. In this chapter, we assume
that the model is completely known, i.e., the parameters ai,j and bk,j are given.
The statistical inference for these parameters is beyond the scope of this chapter.
Observe that if X = x, then (5.1) implies natural equality and inequality con-
straints on the Zj’s. More precisely, (5.1) gives rise to a set of so-called hitting
scenarios. In each hitting scenario, a subset of the Zj’s equal, in other words hit,
their upper bounds and the rest of the Zj’s can take arbitrary values in certain open
intervals. We will show that the regular conditional probability of Z | X = x is
a weighted mixture of the various distributions of the vector Z, under all possible
hitting scenarios corresponding to X = x.
The resulting formula, however, involves determining all hitting scenarios, which
becomes computationally prohibitive for large and even moderate values of p. This
issue is closely related to the NP-hard set-covering problem in computer science (see
e.g. [13]).
Fortunately, further detailed analysis of the probabilistic structure of the max-
linear models allows us to obtain a different formula of the regular conditional prob-
ability (Theorem V.9). It yields an exact and computationally efficient algorithm,
which in practice can handle complex max-linear models with p in the order of thou-
sands, on a conventional desktop computer. The algorithm is implemented in the R
([74]) package maxLinear [107], with the core part written in C/C++. We also used
the R package fields ([37]) to generate some of the figures in this chapter.
We illustrate the performance of our algorithm over two classes of processes: the
max-autoregressive moving average (MARMA) time series (Davis and Resnick [19]),
and the Smith model (Smith [98]) for spatial extremes. The MARMA processes
are spectrally discrete max-stable processes, and our algorithm applies directly. In
61
Section 5.4, we demonstrate the prediction of MARMA processes by conditional
sampling and compare our result to the projection predictors proposed in [19]. To
apply our algorithm to the Smith model, on the other hand, we first need to discretize
the (spectrally continuous) model. Section 5.5 is devoted to conditional sampling
for the discretized Smith model. Thanks to the computational efficiency of our
algorithm, we can choose a mesh fine enough to obtain a satisfactory discretization.
Figure 5.1 shows four realizations from such a discretized Smith model, conditioning
only on 7 observations (with assumed value 5). The algorithm applies in the same
way to more complex models.
−2 −1 0 1
−2−1
01
5
5 5 5
10
10
10
15
15
55
5
5
55
5
−2 −1 0 1
−2−1
01
2
4
4
4
6
6
6
6
8
8
8
8
10
10
12
55
5
5
55
5
−2 −1 0 1
−2−1
01
5
5
5
5 5
10
10
15
55
5
5
55
5
−2 −1 0 1
−2−1
01
5
5
5
5
5
5
10
10
15
55
5
5
55
5
Conditional sampling from the Smith model
Parameters:ρ=0,β1=1,β2=1
5
10
15
Figure 5.1:Four samples from the conditional distribution of the discrete Smith model (see Sec-tion 5.5), given the observed values (all equal to 5) at the locations marked by crosses.
62
Remark V.1. We shall focus on spectrally discrete max-stable processes (see Chap-
ter II):
Xt :=p
j=1
φj(t)Zj, t ∈ T,
where the φj(t)’s are non-negative deterministic functions. By taking sufficiently
large p’s and with judicious φj(t)’s, one can build flexible models that can replicate
the behavior of an arbitrary max-stable process (recall the metric (2.5) characterizing
the convergence of stochastic extremal integrals). From this point of view, a satisfac-
tory computational solution must be able to deal with max-linear models with large
p’s.
Remark V.2. After our work [112] was published, the exact conditional distributions
of spectrally continuous max-stable processes were addressed by Dombry and Eyi–
Minko [31] via a different approach. Nevertheless, they also have a similar notion of
hitting scenarios introduced below.
5.2 Conditional Probability in Max-linear Models
Consider the max-linear model in (5.1). We shall denote this model by:
(5.2) X = A⊙ Z,
where A = (ai,j)n×p is a matrix with non-negative entries, X = (X1, . . . , Xn) and
Z = (Z1, . . . , Zp) are column vectors. We assume that the Zj’s, j = 1, . . . , p, are
independent non-negative random variables having probability densities.
In this section, we provide an explicit formula for the regular conditional probability
of Z with respect to X (see Theorem V.4 below). We start with some intuition and
notation. Throughout this chapter, we assume that the matrix A has at least one
nonzero entry in each of its rows and columns. This will be referred to as Assumption
A.
63
Observe that if x = A⊙ z with x ∈ Rn
+, z ∈ Rp
+, then
(5.3) 0 ≤ zj ≤ zj ≡ zj(A,x) := min1≤i≤n
xi/ai,j, j = 1, . . . , p.
That is, the max-linear model (5.2) imposes certain inequality and equality con-
straints on the Zj’s, given a set of observed Xi’s. Namely, some of the upper bounds
zj(A,x) in (5.3) must be attained, or hit, i.e., zj = zj(A,x) in such a way that
xi = ai,j(i)zj(i), i = 1, . . . , n,
with judicious j(i) ∈ 1, . . . , p. The next example helps to understand the inequality
and equality constraints.
Example V.3. Suppose that n = p = 3 and
A =
1 0 0
1 1 0
1 1 1
.
Let x = A⊙z for some z ∈ R3+. In this case, it necessarily follows that x1 ≤ x2 ≤ x3.
Moreover, (5.3) yields z = x.
(i) If x = (1, 2, 3), then it trivially follows that z = z = (1, 2, 3), which is an
equality constraint on z.
(ii) If x = (1, 1, 3), then it follows that z1 = z1 = 1, z2 ≤ z2 = 1 and z3 = z3 = 3.
Here, the “equality constraints” must hold for z1 = z1 and z3 = z3, while z2
only needs to satisfy the “inequality constraint” 0 ≤ z2 ≤ z2.
Write
C(A,x) := z ∈ Rp
+ : x = A⊙ z,
64
and note that the conditional distribution of Z | X = x concentrates on the set
C(A,x). The observation in Example V.3 can be generalized and formulated as
follows.
• Every z ∈ C(A,x) corresponds to a set of active (equality) constraints J ⊂
1, . . . , p, which we refer to as a hitting scenario of (A,x), such that
(5.4) zj = zj(A,x), j ∈ J and zj < zj(A,x), j ∈ J c := 1, . . . , p \ J.
Observe that if j ∈ J , then there are no further constraints and zj can take any
value in [0, zj), regardless of the values of the other components of the vector
z ∈ C(A,x).
• Every value x may give rise to many different hitting scenarios J ⊂ 1, . . . , p.
Let J (A,x) denote the collection of all such J ’s. We refer to J (A,x) as to the
hitting distribution of x w.r.t. A:
J (A,x) ≡J ⊂ 1, . . . , p : exist z ∈ C(A,x), such that (5.4) holds
.
To illustrate the notions of hitting scenario and hitting distribution, consider again
Example V.3. Therein, we have J (A,x) = 1, 2, 3 in case (i), and J (A,x) =
1, 3, 1, 2, 3 in case (ii).
The hitting distribution J (A,x) is a finite set and thus can always be identified.
However, the identification procedure is the key difficulty in providing an efficient
algorithm for conditional sampling in practice. This issue is addressed in Section 5.3.
In the rest of this section, suppose that J (A,x) is given. Then, we can partition
C(A,x) as follows
C(A,x) =
J∈J (A,x)
CJ(A,x) ,
65
where
CJ(A,x) = z ∈ Rp
+ : zj = zj, j ∈ J and zj < zj, j ∈ J.
The sets CJ(A,x), J ∈ J (A,x) are disjoint since they correspond to different hitting
scenarios in J (A,x). Let
(5.5) r(J (A,x)) = minJ∈J (A,x)
|J | ,
where |J | is the number of elements in J . We call r(J (A,x)) the rank of the hitting
distribution J (A,x). It equals the minimal number of equality constraints among the
hitting scenarios in J (A,x). It will turn out that the hitting scenarios J ⊂ J (A,x)
with |J | > r(J (A,x)) occur with (conditional) probability zero and can be ignored.
We therefore focus on the set of all relevant hitting scenarios:
Jr(A,x) = J ∈ J (A,x) : |J | = r(J (A,x)).
Theorem V.4. Consider the max-linear model in (5.2), where Zj’s are independent
random variables with densities fZjand distribution functions FZj
, j = 1, . . . , p. Let
A = (ai,j)n×p have non-negative entries satisfying Assumption A and let RRp
+be the
class of all rectangles (e, f ], e, f ∈ Rp
+ in Rp
+.
For all J ∈ J (A,x), E ∈ RRp
+, and x ∈ R
n
+, define
(5.6) νJ(x, E) :=
j∈J
δzj(πj(E))
j∈Jc
PZj ∈ πj(E) | Zj < zj,
where πj(z1, . . . , zp) = zj and δa is a unit point-mass at a.
Then, the regular conditional probability ν(x, E) of Z w.r.t. X equals:
(5.7) ν(x, E) =
J∈Jr(A,x)
pJ(A,x)νJ(x, E), E ∈ RRp
+,
for PX-almost all x ∈ A⊙ (Rp
+), where for all J ∈ Jr(A,x),
(5.8) pJ(A,x) =wJ
K∈Jr(A,x) wK
with wJ =
j∈J
zjfZj(zj)
j∈Jc
FZj(zj).
66
In the special case when the Zj’s are α-Frechet with scale coefficient 1, we have
wJ =
j∈J(zj)−α.
Remark V.5. We state (5.7) only for rectangle sets E because the projections πj(B)
of an arbitrary Borel set B ⊂ Rp
+ are not always Borel (see e.g. [99]). Nevertheless,
the extension of measure theorem ensures that Formula (5.7) specifies completely
the regular conditional probability.
We do not provide a proof of Theorem V.4 directly. Instead, we will first provide
an equivalent formula for ν(x, E) in Theorem V.9 in Section 5.3, and then prove that
ν(x, E) is the desired regular conditional probability. All the proofs are deferred to
Section 5.6. The next example gives the intuition behind Formula (5.7).
Example V.6. Continue with Example V.3.
(i) IfX = x = (1, 2, 3), then z = x, J (A,x) = 1, 2, 3. Therefore, r(J (A,x)) =
We discuss here important computational issues related to sampling from the reg-
ular conditional probability in (5.7). It turns out that identifying all hitting scenarios
amounts to solving the set covering problem, which is NP-hard (see e.g. [13]). The
probabilistic structure of the max-linear models, however, will lead us to an alter-
native efficient solution, valid with probability one. In particular, we will provide
a new formula for the regular conditional probability, showing that Z can be de-
composed into conditionally independent vectors, given X = x. As a consequence,
with probability one we are not in the ‘bad’ situation that the corresponding set
covering problem requires exponential time to solve. Indeed, this will lead us to an
efficient and linearly-scalable algorithm for conditional sampling, which works well
for max-linear models with large dimensions n× p arising in applications.
To fix ideas, observe that Theorem V.4 implies the following simple algorithm.
Algorithm I:
1. Compute zj for j = 1, . . . , p.
2. Identify J (A,x), compute r = r(J (A,x)) and focus on the set of relevant
69
hitting scenarios Jr = Jr(A,x).
3. Compute wJJ∈Jrand pJJ∈Jr
.
4. Sample Z ∼ ν(x, ·) according to (5.7).
Step 1 is immediate. Provided that Step 2 is done, Step 3 is trivial and, Step 4
can be carried out by first picking a hitting scenario J ∈ Jr(A,x) (with probability
pJ(A,x)), setting Zj = zj, for j ∈ J and then resampling independently the remain-
ing Zj’s from the truncated distributions: Zj | Zj < zj, for all j ∈ 1, . . . , p \ J .
The most computationally intensive aspect of this algorithm is to identify the
set of all relevant hitting scenarios Jr(A,x) in Step 2. This is closely related to
the NP-hard set covering problem in theoretical computer science (see e.g. [13]),
which is formulated next. Let H = (hi,j)n×p be a matrix of 0’s and 1’s, and let
c = (cj)p
j=1 ∈ Zp
+ be a p-dimensional cost vector. For simplicity, introduce the
notation:
m ≡ 1, 2, . . . ,m, m ∈ N.
For the matrix H, we say that the column j ∈ p covers the row i ∈ n, if hi,j = 1.
The goal of the set-covering problem is to find a minimum-cost subset J ⊂ p, such
that every row is covered by at least one column j ∈ J . This is equivalent to solving
(5.14) minδj∈0,1j∈p
j∈p
cjδj , subject to
j∈p
hi,jδj ≥ 1 , i ∈ n .
We can relate the problem of identifying Jr(A,x) to the set covering problem by
defining
(5.15) hi,j = 1ai,jzj=xi,
where A = (ai,j)n×p and x = (xi)ni=1 are as in (5.2), and cj = 1 , j ∈ p. It is easy
70
to see that, every J ∈ Jr(A,x) corresponds to a solution of (5.14), and vice versa.
Namely, for δjj∈p minimizing (5.14), we have J = j ∈ p : δj = 1 ∈ Jr(A,x).
The set Jr(A,x) corresponds to the set of all solutions of (5.14), which depends
only on the matrix H. Therefore, in the sequel we write Jr(H) for Jr(A,x), and
(5.16) H = (hi,j)n×p ≡ H(A,x),
with hi,j as in (5.15) will be referred to as the hitting matrix.
Example V.8. Recall Example V.6. The following hitting matrices correspond to
the three cases of x discussed therein:
H(i) =
1 0 0
0 1 0
0 0 1
, H(ii) =
1 0 0
1 1 0
0 0 1
and H(iii) =
1 0 0
1 1 0
1 1 1
.
Observe that solving for Jr(H) is even more challenging than solving the set
covering problem (5.14), where only one minimum-cost subset J is needed, and
often an approximation of the optimal solution is acceptable. Here, we need to
identify exhaustively all J ’s such that (5.14) holds. Fortunately, this problem can
be substantially simplified, thanks to the probabilistic structure of the max-linear
model.
We first study the distribution ofH. In view of (5.16), we have thatH = H(A,X),
with X = A⊙Z, is a random matrix. It will turn out that, with probability one, H
has a nice structure, leading to an efficient conditional sampling algorithm.
For any hitting matrix H, we will decompose the set p ≡ 1, . . . , p into a
certain disjoint union p =
r
s=1 J(s). The vectors (Zj)
j∈J(s) , s = 1, . . . , r will turn
out to be conditionally independent (in s), given X = x. Therefore, ν(x, E) will be
expressed as a product of (conditional) probabilities.
71
We start by decomposing the set n ≡ 1, . . . , n. First, for all i1, i2 ∈ n , j ∈
p, we write i1j
∼ i2 , if hi1,j = hi2,j = 1. Then, we define an equivalence relation on
n:
(5.17) i1 ∼ i2, if i1 = i0j1∼ i1
j2∼ · · ·
jm∼ im = i2 ,
with some m ≤ n, i1 = i0,i1, . . . ,im = i2 ∈ n, j1, . . . , jm ∈ p. That is, ‘∼’ is the
transitive closure of ‘j
∼’. Consequently, we obtain a partition of n, denoted by
(5.18) n =r
s=1
Is ,
where Is, s = 1, . . . , r are the equivalence classes w.r.t. (5.17). Based on (5.18), we
define further
J (s) =j ∈ p : hi,j = 1 for all i ∈ Is
,(5.19)
J(s)
=j ∈ p : hi,j = 1 for some i ∈ Is
.(5.20)
The sets J (s), J(s)s∈r will determine the factorization form of ν(x, E).
Theorem V.9. Let Z be as in Theorem V.4. Let also H be the hitting matrix
corresponding to (A,X) with X = A ⊙ Z, and J (s), J(s)s∈r be the sets defined
in (5.19) and (5.20). Then, with probability one, we have
(i) r = r(J (A,X)),
(ii) for all J ⊂ p, J ∈ Jr(A,A⊙ Z) if and only if J can be written as
(5.21) J = j1, . . . , jr with js ∈ J (s) , s ∈ r ,
(iii) for ν(x, E) defined in (5.7),
(5.22) ν(X, E) =r
s=1
ν(s)(X, E) with ν(s)(X, E) =
j∈J(s) w
(s)j(X)ν(s)
j(X, E)
j∈J(s) w
(s)j(X)
,
72
where for all j ∈ J (s),
w(s)j(x) := zjfZj
(zj)
k∈J(s)
\j
FZk(zk) ,(5.23)
ν(s)j(x, E) := δπj(E)(zj)
k∈J(s)
\j
P(Zk ∈ πk(E)|Zk < zk),(5.24)
with zj = zj(x) as in (5.3).
The proof of Theorem V.9 is given in Section 5.6.
Remark V.10. Note that this result does not claim that ν(x, E) in (5.22) is the regular
conditional probability. It merely provides an equivalent expression for (5.7), which
is valid with probability one. We still need to show that (5.7), or equivalently (5.22),
is indeed the regular conditional probability.
From (5.23) and (5.24), one can see that ν(s) is the conditional distribution of
(Zj)j∈J
(s) . Therefore, Relation (5.22) implies that (Zj)j∈J
(s)s∈r, as vectors in-
dexed by s, are conditionally independent, given X = x. This leads to the following
improved conditional sampling algorithm:
Algorithm II:
1. Compute zj for j = 1, . . . , p and the hitting matrix H = H(A,x).
2. Identify J (s), J(s)s∈r by (5.19) and (5.20).
3. Compute w(s)jj∈J(s) for all s ∈ r by (5.23).
4. Sample (Zj)j∈J
(s) | X = x ∼ ν(s)(x, ·) independently for s = 1, . . . , r.
5. Combine the sampled (Zj)j∈J
(s) , s = 1, . . . , r to obtain a sample Z.
This algorithm identifies all hitting scenarios in an efficient way. To illustrate
its efficiency compared to Algorithm I, consider that r = 10 and |J (s)| = 10 for all
73
Table 5.1:Means and standard deviations (in parentheses) of the running times (in seconds) forthe decomposition of the hitting matrix H, based on 100 independent observations X =A⊙ Z, where A is an (n× p) matrix corresponding to a discretized Smith model.
Table 5.3 also shows the widths of the upper 95%-confidence intervals. Note that
these widths are not equal to the upper confidence bounds, given by the conditional
95%-quantiles, since the left end-point of the conditional distributions are greater
than zero. When the time lag is small, the left end-point is large and the widths
are small, due to the strong influence of the past of the process Xt100t=1. On the
other hand, because of the weak temporal dependence of the MAR(3) processes, this
influence decreases fast as the lags increase. Consequently, the conditional distri-
bution converges to the unconditional one, and the conditional quantile to the un-
conditional one. Note that the (unconditional) 95%-quantile of Xs for the MARMA
process (5.26) can be calculated via the formula 0.95 = P(σZ ≤ u) = exp(−σu−1),
with σ =
p
j=0 ψj. For the MAR(3) process we chose, we have σ = 3.4 and the
95%-quantile of Xs equals 66.29. This is consistent with the widths in Table 5.3 for
large lags.
Remark V.12. As pointed out by an anonymous referee, in this case one can directly
generate samples from XsN
s=n+1 | Xtn
t=1, by generating independent Frechet ran-
dom variables and iterating (5.33). We selected this example only for illustrative
purpose and to be able to compare with the projection predictors in [19]. One can
modify slightly the prediction problem, such that our algorithm still applies by ad-
justing accordingly (5.30), while both the projection predictor and the direct method
by using (5.33) do not apply. For example, consider the prediction problem with re-
spect to the conditional distribution P(Xs2n+N
s=2n+1 ∈ · | Xt : t = 1, 3, . . . , 2n − 1)
(prediction with only partial history observed) or P(Xsn−1s=2 ∈ · | X1, Xn) (predic-
tion of the middle path with the beginning and the end-point (in the future) given).
In other words, our algorithm has no restriction on the locations of observations.
This feature is of great importance in spatial prediction problems.
80
5.5 Discrete Smith Model
Consider the following moving maxima random field model in R2:
(5.36) Xt =
e
R2φ(t− u)Mα(du), t = (t1, t2) ∈ R
2,
where Mα is an α-Frechet random sup-measure on R2 with the Lebesgue control
measure. Smith [98] proposed to use for φ the bivariate Gaussian density:
(5.37) φ(t1, t2) :=β1β2
2π1− ρ2
exp−
1
2(1− ρ2)
β21t
21 − 2ρβ1β2t1t2 + β2
2t22
,
with correlation ρ ∈ (−1, 1) and variances σ2i= 1/β2
i, i = 1, 2. Consistent and
asymptotically normal estimators for the parameters ρ, β1 and β2 were obtained by
de Haan and Pereira [25]. Here, we will assume that these parameters are known
and will illustrate the conditional sampling methodology over a discretized version
of the random field (5.36). Namely, we truncate the extremal integral in (5.36) to
the square region [−M,M ]2 and consider a uniform mesh of size h := M/q, q ∈ N.
We then set
(5.38) Xt :=
−q≤j1,j2≤q−1
h2/αφ(t− uj1j2)Zj1j2 ,
where uj1j2 = ((j1 + 1/2)h, (j2 + 1/2)h) and h2/αZj1j2
d= Mα((j1h, (j1 + 1)h] ×
(j2h, (j2 + 1)h]). This discretized model (5.38) can be made arbitrarily close to
the spectrally continuous one in (5.36) by taking a fine mesh h and sufficiently large
M (see e.g. [101]).
Suppose that the random fieldX in (5.38) is observed at n locationsXti = xi, ti ∈
[−M,M ]2, i = 1, . . . , n. In view of (5.38), we have the max-linear model X = A⊙Z,
with X = (Xti)n
i=1 and Z = (Zj)p
j=1, p = q2. By sampling from the conditional
distribution of Z | X = x, we can predict the random field Xs at arbitrary locations
s ∈ R2.
81
To illustrate our algorithm, we used the model (5.38) with parameter values ρ =
0, β1 = β2 = 1,M = 4, p = q2 = 2500, and n = 7 observed locations. We generated
N = 500 independent samples from the conditional distribution of the random field
Xs, where s takes values on an uniform 100×100 grid, in the region [−2, 2]×[−2, 2].
We have already seen four of these realizations in Figure 5.1. Figure 5.3 illustrates
the median and 0.95-th quantile of the conditional distribution. The former provides
the optimal predictor for the values of the random field given the observed data, with
respect to the absolute deviation loss. The marginal quantiles, on the other hand,
provide important confidence regions for the random field, given the data.
Certainly, conditional sampling may be used to address more complex functional
prediction problems. In particular, given a two-dimensional threshold surface, one
can readily obtain the correct probability that the random field exceeds or stays
below this surface, conditionally on the observed values. This is much more than
what marginal conditional distributions can provide.
−2 −1 0 1
−2−1
01
4.0
4.5
5.0
5.5
6.0
4
4.5
5
5
5
5.5
5.5 5.5
5.5
5.5
5.5
5.5
6
6
6
6
Conditional Median of the Smith model
Parameters:ρ=0, β1=1, β2=1
5
5
5
5
5
5
5
−2 −1 0 1
−2−1
01
5
10
15
20
25
10 15
15
15
15
20
20
20
20
25
25 25
Conditional Marginal Quantile of the Smith model
Parameters:ρ=0, β1=1, β2=1, q=0.95
5
5
5
5
5
5
5
Figure 5.3:Conditional medians (left) and 0.95-th conditional marginal quantiles (right). Eachcross indicates an observed location of the random field, with the observed value atright.
82
5.6 Proofs of Theorems V.4 and V.9
In this section, we prove Theorems V.4 and V.9. We will first prove Theorem V.9,
which simplifies the regular conditional probability formula (5.7) in Theorem V.4.
Then, we show the simplified new formula is the desired regular conditional probabil-
ity, which completes the proof of Theorem V.4. The key step to prove Theorem V.9
is the following lemma. Write H·j = i ∈ r : hi,j = 1.
Lemma V.13. Under the assumptions of Theorem V.9, with probability one,
(i) J (s)is nonempty for all s ∈ r, and
(ii) for all j ∈ J (s), H·j ∩ Is = ∅ implies H·j ⊂ Is.
Proof. Note that to show part (ii) of Lemma V.13, it suffices to observe that since
Is is an equivalence class w.r.t. Relation (5.17), H·j \ Is and H·j ∩ Is cannot be both
nonempty. Thus, it remains to show part (i). We proceed by excluding several
P-measure zero sets, on which the desired results may not hold.
First, observe that for all i ∈ n, the maximum value of ai,jZjj∈r is achieved
for unique j ∈ p with probability one, since the Zj’s are independent and have
continuous distributions. Thus, the set
N1 :=
i∈n,j1,j2∈p,j1 =j2
ai,j1Zj1 = ai,j2Zj2 = max
j∈p
ai,jZj
has P-measure zero. From now on, we focus on the event N c
1 and set j(i) =
argmaxj∈pai,jZj for all i ∈ n.
Next, we show that with probability one, i1j
∼ i2 implies j(i1) = j(i2). That is,
the set
N2 :=
j∈p,i1,i2∈n,i1 =i2
Nj,i1,i2 with Nj,i1,i2:=
j(i1) = j(i2), i1
j
∼ i2
83
has P-measure 0. It suffices to show P(Nj,i1,i2) = 0 for all i1 = i2. If not, since p
and n are finite sets, there exists N0 ⊂ Nj,i1,j2 , such that j(i1) = j1 = j(i2) = j2
on N0, and P(N0) > 0. At the same time, however, observe that i1j
∼ i2 implies
hi1,j = hi2,j = 1, which yields
aik,jzj = xik= aik,j(ik)Zj(ik) = aik,jkZjk
, k = 1, 2 .
It then follows that on N0, Zj1/Zj2 = ai1,jai2,j2/(ai2,jai1,j1), which is a constant. This
constant is strictly positive and finite. Indeed, this is because on N c
1 , ai,j(i) > 0 by
Assumption A and hi,j = 1 implies ai,j > 0. Since Zj1 and Zj2 are independent
continuous random variables, it then follows that P(N0) = 0.
Finally, we focus on the event (N1∪N2)c. Then, for any i1, i2 ∈ Is, we have i1 ∼ i2
and let i0, . . . ,in be as in (5.17). It then follows that j(i1) = j(i0) = j(i1) = · · · =
j(in) = j(i2). Note that for all i ∈ n, hi,j(i) = 1 by the definition of j(i). Hence,
j(i1) = j(i2) ∈ J (s). We have thus completed the proof.
Proof of Theorem V.9. Since Iss∈r are disjoint with
s∈rIs = n, in the lan-
guage of the set-covering problem, to cover n, we need to cover each Is. By part
(ii) of Lemma V.13, any two different Is1 and Is2 cannot be covered by a single set
H·j. Thus we need at least r sets to cover n. On the other hand, with probability
one we can select one js from each J (s) (by part (i) of Lemma V.13), which yields a
valid cover. That is, with probability one, r = r(J (H)) and any valid minimum-cost
cover of n must be as in (5.21), and vice versa. We have thus proved parts (i) and
(ii).
84
To show (iii), by straight-forward calculation, we have, with probability one,
J∈Jr(A,x)
wJ =
j1∈J(1)
· · ·
jr∈J(r)
wj1,...,jr
=
j1∈J(1)
· · ·
jr−1∈J(r−1)
r−1
s=1
zjsfZjs(zjs)
j /∈J(r)
j =j1,...,jr−1
FZj(zj)
×
j∈J(r)
zjfZj
(zj)
k∈J(r)
\j
FZk(zk)
=r
s=1
j∈J(s)
zjfZj
(zj)
k∈J(s)
\j
FZk(zk)
=
r
s=1
j∈J(s)
w(s)j
.(5.39)
Similarly, we have
(5.40)
J∈Jr(A,x)
wJνJ(x, E) =r
s=1
j∈J(s)
w(s)jν(s)j(x, E)
.
By plugging (5.39) and (5.40) into (5.7), we obtain the desired result and complete
the proof.
Proof of Theorem V.4. To prove that ν in (5.7) yields the regular conditional prob-
ability of Z given X, it is enough to show that
(5.41) P(X ∈ D,Z ∈ E) =
D
ν(x, E)PX(dx),
for all rectangles D ∈ RRn
+and E ∈ RR
p
+. In view of Theorem V.9, it is enough to
work with ν(x, E) given by (5.22).
We shall prove (5.41) by breaking the integration into a suitable sum of integrals
over regions corresponding to all hitting matrices H for the max-linear model X =
A⊙ Z. We say such a hitting matrix H is nice, if J (s) defined in (5.19) is nonempty
for all s ∈ r. In view of Lemma V.13, it suffices to focus on the set H(A) of
nice hitting matrices H. Notice that the set H(A) is finite since the elements of the
hitting matrices are 0’s and 1’s.
85
For all rectangles D ∈ RRn
+, let
DH =x = A⊙ z : H(A,x) = H,x ∈ D
be the set of all x ∈ Rn
+ that give rise to the hitting matrix H. By Lemma V.13 (i),
for the random vector X = A⊙ Z, with probability one, we have
X =
H∈H(A)
X1DH(X)
and hence
(5.42)
D
ν(x, E)PX(dx) =
H∈H(A)
DH
ν(x, E)PX(dx) .
Now fix an arbitrary and non-random nice hitting matrix H ∈ H(A). Let Iss∈r
denote the partition of n determined by (5.17) and let J (s), J(s), s = 1, . . . , r be
as in (5.19). Recall that J (s) ⊂ J(s)
and the sets J(s), s = 1, . . . , r are disjoint.
Focus on the set DH ⊂ Rn
+. Without loss of generality, and for notational conve-
nience, suppose that s ∈ Is, for all s = 1, . . . , r. That is,
sufficient condition such that Condition VII.3 holds is
(7.14) γ <dβ
d+ βand β > d .
Consequently, if E(|0|α) < ∞ for some α > 2, and Condition VII.1 and (7.14) hold,
then the asymptotic normality (7.3) follows.
Proof. Assume that mn takes the form of nδ. Observe that Bmnis of the same
order of A[mn] as n → ∞. Then, the limit conditions (7.9), (7.10) and (7.11) are
implied by
limn→∞
n−βδ+γ + ndδ−γ + nδ−1+γ/d = 0 ,
which is equivalent to γ/β < δ < minγ/d, 1 − γ/d. Since β > d implies that
∆∞ < ∞, the desired result follows.
Remark VII.7. Under the assumptions of Corollary VII.6, Condition (7.14) is very
close to necessary for Condition VII.3 to holds. Indeed, if A[n] = l(n)n−β with
limn→∞ l(n) = c2 > 0, then the same argument above yields that Condition VII.3 is
equivalent to (7.14).
Below, we provide examples of coefficients so that Condition VII.3 holds. We
assume that bn = n−γ for some γ ∈ (0, d).
122
Example VII.8. We compare our conditions and the ones by Hallin et al. [44].
They considered the case that |ai| ≤ C|i|−q
∞, i 0. Then, they require
(7.15) q > max(d+ 3, 2d+ 1/2) and limn→∞
ndb(2q−1+6d)/(2q−1−4d)n
= ∞ .
Our condition (7.14) imposes weaker assumption in this case (with bn = n−γ). First,
observe that
A2n,1,...,1 ≤ B2
n≤ C
∞
i=n
id−1i−2q≤ Cnd−2q .
We can apply Corollary VII.6 with β = q − d/2. Then, (7.14) becomes
(7.16) q >3d
2and γ < d
q − d/2
q + d/2.
Thus, to establish the asymptotic normality (7.3), our condition (7.16) is less restric-
tive than (7.15) on both q and γ.
Example VII.9. We compare our conditions and the ones by El Machkouri [34].
Note that his results apply to general stationary random fields and the linear random
fields are a specific case. In particular, he showed that for causal linear random fields,
if
(7.17)
i∈Zd
|i|q∞|ai| < ∞
with q = 5d/2, then the asymptotic normality follows.
In this case, our condition on the coefficients is weaker, requiring only q > d.
Indeed, suppose (7.17) holds with some q > 0. Then, to apply Corollary VII.6, it
suffices to observe
A2n,1 =
∞
i1=n
i2,...,id∈N
|ai|2≤ Cn−2q
∞
i1=n
i2,...,id∈N
|i|2q∞|ai|
2 < Cn−2q,
and take β = q.
123
At the same time, our result requires γ < dq/(q+d) for the bandwidth, in addition
to the minimal one (7.2) assumed in [34]. Recall also that we assume E(|0|α) < ∞ for
some α > 2, while El Machkouri’s result needs only finite-second-moment assumption
on 0.
Remark VII.10. Finally, we compare our result to Wu and Mielniczuk [120]. In the
one-dimensional case, to have asymptotic normality they assume only finite variance
of 0 and weaker assumption on the coefficient:
(7.18)∞
i=0
|ai| < ∞ .
This is weaker than our condition in one dimension (with q > d = 1 in (7.17)).
Wu and Mielniczuk followed a martingale approximation approach. It remains an
open question that in high dimension, whether the condition q > d in (7.17) can be
improved to match (7.18) in dimension one.
7.3 A Central Limit Theorem for m-Dependent Random Fields
In this section, we prove a central limit theorem for stationary triangular arrays of
m-dependent random fields. Throughout this section, let Yn,i : i ∈ Ndn∈N denote
stationary zero-mean triangular arrays. That is, for each n, Yn,ii∈Nd is stationary
and Yn,i has zero mean. Furthermore, we assume that Yn,ii∈Nd is mn-dependent in
the sense that Yn,i and Yn,j are independent if |i− j|∞ ≥ m. We provide conditions
such that
(7.19)Sn(Y )
nd/2≡
i∈1,nd Yn,i
nd/2⇒ N (0, σ2) as n → ∞.
A key condition is the following:
(7.20)
i∈Nd,1ij
Yn,i
2≤ C(j1 · · · jd)
1/2 for all n ∈ N, j ∈ Nd .
124
Remark VII.11. Observe that Proposition VI.20 provides conditions such that (7.20)
holds. In fact, inequality (7.20) has been established, under various conditions on
the dependence of stationary random fields, by Dedecker [28] and El Machkouri et
al. [35], among others.
Theorem VII.12. Suppose that there exists a constant C such that (7.20) holds. If
there exists a sequence lnn∈N ⊂ N, mn/ln → 0 and ln/n → 0 as n → ∞, such that
limn→∞
1
ldn
E
k∈1,lndYn,k
2= σ2 ,(7.21)
limn→∞
1
ldn
E
k∈1,lndYn,k
21
k∈1,lndYn,k
> nd/2
= 0 ,(7.22)
for all > 0, then (7.19) holds.
Proof. Consider partial sums over big blocks of size ldn, denoted by
ηn,k =
i∈1,lndYn,i+k(ln+mn) , k ∈ N
d.
In this way, for each n ∈ N, ηn,kk∈Nd are i.i.d., as we separate neighboring blocks
by distance mn, and Yn,ii∈Zd are mn-dependent. Set
Sn(η) =
k∈0,n/(ln+mn)−1dηn,k , n ∈ N .
Then, (7.20) implies that
Sn(Y )
nd/2−
Sn(η)
nd/2
2→ 0 as n → ∞ .
To see this, for the sake of simplicity, we consider the case n/(ln + mn) =
n/(ln +mn). Indeed, by the triangular inequality, the left-hand side above can
be bounded by sums in form of
i∈BYn,i
2/nd/2, where B can be a rectangle of
size nd−rmr
nwith r ∈ 1, . . . , d − 1. Focusing on the dominant term with r = 1,
125
we then bound the left-hand side above by C(n/(ln + mn))1/2(nd−1mn)1/2/nd/2 =
Cm1/2n /(ln +mn)1/2 → ∞ as n → ∞.
As a consequence, it suffices to show Sn(η)/nd/2 ⇒ N (0, σ2). This, under condi-
tions (7.21) and (7.22), follows from the standard central limit theorem for triangular
arrays of independent random variables (see e.g. [32], Chapter 2, Theorem 4.5).
Remark VII.13. Central limit theorems for mn-dependent random fields has been
considered by Heinrich [47]. His result has been recently applied, with mn = m
fixed, by El Machkouri et al. [35] to establish a central limit theorem for stationary
random fields.
Our application requires us to take mn → ∞. In this case our condition in
Theorem VII.12 is weaker than Heinrich’s. In particular, he assumed
(7.23) limn→∞
m2dn
nd
i∈1,ndE
Y 2n,i1|Yn,i|>nd/2m
−2dn
= 0 , for all > 0 .
This is stronger than (7.22).
7.4 Asymptotic Normality by m-Approximation
In this section, we prove Theorem VII.4 by an m-approximation argument. Fix
x ∈ R and write
Zn,i =1
√bnKx−Xi
bn
and ζn,i =
1√bnKx−Xi,mn
bn
, i ∈ Z
d .
In this way, ζn,ii∈Zd are mn-dependent. We will use ζn,i : i ∈ Zdn∈N to approx-
imate Zn,i : i ∈ Zdn∈N. We also write Zn,i = Zn,i − EZn,i and ζ
n,i= ζn,i − Eζn,i.
Setting
Sn(ζ) =
i∈1,ndζn,i
and Sn(Z − ζ) =
i∈1,nd(Zn,i − ζ
n,i) ,
126
we decompose
(7.24) (ndbn)1/2(fn(x)− Efn(x)) =
Sn(ζ)
nd/2+
Sn(Z − ζ)
nd/2.
To prove Theorem VII.4, it suffices to establish the following two results.
Proposition VII.14. Under Condition VII.1 and (7.8), (7.10), (7.11) of Condi-
tion VII.3,
(7.25)Sn(ζ)
nd/2⇒ N (0, σ2
x) .
Proposition VII.15. Under Condition VII.1 and (7.8), (7.9) of Condition VII.3,
(7.26)Sn(Z − ζ)
nd/2
P−→ 0 .
To prove the above two propositions, a key step is to establish the following
moment inequalities.
Lemma VII.16. There exists a constant C > 0, such that for all n ∈ N,
(7.27)Sn(Z − ζ)
2≤ Cnd/2
Zn,0 − ζn,0
2+ b1/2
n∆n
.
In addition, if E(|0|α) < ∞ for some α ≥ 2, then
(7.28)
i∈Nd,1ij
ζn,i
α
≤ C(j1 · · · jd)1/2
ζn,0
α+ b1/2
n∆n
, for all j ∈ N
d.
The proof is deferred to Section 7.5.
Proof of Proposition VII.14. Observing that Sn(ζ)/nd/2 is a partial sum of mn-
dependent random fields, we apply Theorem VII.12. Observe that sinceζ
n,0
2→ σx
as n → ∞, (7.28) with α = 2 and assumption (7.8) entail (7.20). Thus, to
prove (7.25), it suffices to show, for ln = mn log n,
(7.29) limn→∞
1
ldn
E
i∈1,lndζn,i
2= σ2
x,
127
and, writing ξn =
i∈1,lnd ζn,i,
(7.30) limn→∞
1
ldn
E
ξ2n1|ξn|>nd/2
= 0 , for all > 0 .
By standard calculation, under (7.7) of Condition VII.1, for all n ∈ N and i = 0,
|E(ζn,0ζn,i)| ≤ Cp
i,mnbn ≤ Cbn .
Therefore,
1
ldn
E
i∈1,lndζn,i
2− Eζ
2n,0
≤ 2
i∈−mn,mnd|E(ζ
n,0ζn,i)|1i =0 ≤ Cmd
nbn .
Thus, assumption (7.10) entails (7.29). To prove (7.30), observe that
E(ξ2n1|ξn|>nd/2) ≤ ξn
2αP(|ξn| > nd/2)(α−2)/α
≤ ξn2α
ξn
22
nd2
(α−2)/α.
This time, (7.28) and (7.8) yield ξn2 ≤ Cld/2n . For α > 2, observe that, since K is
bounded,
ζn,0
α= (E|ζ
n,0|α)1/α ≤
C
b(α−2)/2n
ζn,0
2
2
1/α≤ Cb−(α−2)/(2α)
n.
So, ξn2α≤ Cld
nb−(α−2)/αn . To sum up, we have obtained that
1
ldn
E(ξ2n1|ξn|>nd/2) ≤ C
ldn
ndbn
(α−2)/α.
Now, (7.11) entails (7.30).
Proof of Proposition VII.15. In order to obtain the desired result, it suffices to com-
bine (7.27), assumptions (7.8) and (7.9) and Lemma VII.17 below.
Lemma VII.17. Under the assumption of Condition VII.1, there exists a constant
C, such that for all n ∈ N,
(7.31)ζ
n,0 − Zn,0
2≤ C
Bmn
bn
1/2+ b1/2
n
.
The proof is deferred to Section 7.5.
128
7.5 Proofs
Proof of Lemma VII.2. (i) The existence and Lipschitz continuity of p and pm have
been proved by Wu and Mielniczuk [120], Lemma 1. To prove (7.6), observe that
|pm(y)− p(y)| ≤
|pm(y)− pm(y − x)|pm(x)dx
≤ C
|x|pm(x)dx = CE| X0,m| .(7.32)
This entails that pm(x) → p(x) uniformly for x ∈ R as m → ∞. Therefore, (7.6)
holds.
(ii) Fix i ∈ Zd \ 0 and let Fi denote the joint distribution function of (X0, Xi).
For the sake of simplicity, we prove the case of a0 = 1. Write R = X0 − 0 and
Ri = Xi− i− ai0. Now, R and Ri are dependent random variables. First, we show
that
(7.33) pi(x, y) ≡∂2
∂x∂yFi(x, y) = E[p(x−R)p(y −Ri − aix)] .
Indeed,
Fi(x, y) = P(X0 ≤ x,Xi ≤ y)
= P(0 +R ≤ x, i + ai0 +Ri ≤ y)
= EΦi(x−R, y −Ri) ,(7.34)
with, letting F denote the cumulative distribution function of 0,
Φi(x, y) =
x
−∞
F(y − aix)F(dx
) .
Differentiating (7.34) yields (7.33) (see e.g. [32], Appendix A.9 on the validation of
exchange of differentiation and expectation).
129
Next, we prove (7.7) by establishing the following two steps:
(7.35) lim|i|∞→∞
supx,y
|pi(x, y)− p(x)p(y − aix)| = 0 ,
and
(7.36) limm→∞
supx,y,i
|pi(x, y)− pi,m(x, y)| = 0 .
Then, (7.35) implies the first part of (7.7), and the two limits imply the second part.
To prove (7.35), set
Di = E(Ri | σ(k : k 0)) and Di = Ri −Di , i ∈ Z
d.
By definition, Di and R are independent. Introducing an intermediate term E[p(x−
R)p(y−Di−aix)] = p(x)Ep(y−Di−aix), we then bound |pi(x, y)−p(x)p(y−aix)| ≤
Ψ1 +Ψ2 with, under the assumption that p is bounded and Lipschitz,
Ψ1 = |pi(x, y)− E[p(x−R)p(y −Di − aix)]|
≤ E[p(x−R)|Ri −Di|] ≤ CE| Di|,
and
Ψ2 = |p(x)p(y − aix)− E[p(x−R)p(y −Di − aix)]|
≤ p(x)E|p(y − aix−Ri + ai0)− p(y −Di − aix)|
≤ C(E| Di|+ |ai|) .
By (7.13), |pi(x, y)− p(x)p(y − aix)| → 0 as |i|∞ → ∞.
To prove (7.36), define Rm = X0,m − 0 and Ri,m = Xi,m − i − ai01|i|∞<m.
Then, similarly as (7.33), one has
pi,m(x, y) = E[p(x−Rm)p(y − aix1|i|∞<m −Ri,m)] .
130
Introducing an intermediate term E[p(x−R)p(y− aix1|i|∞<m −Ri,m)], we obtain
that
|pi,m(x, y)− pi(x, y)|
≤ E[p(x−R)(|aix|1|i|∞≥m + |Ri −Ri,m|)] + CE|R−Rm|
≤ C(|x|p(x)|ai|1|i|∞≥m + |R−Rm|+ |Ri −Ri,m|) .
Since X0 has finite second moment and p is bounded and Lipschitz, supx|x|p(x) <
∞. The summability assumption on ai implies that limm→∞ sup|i|∞≥m|ai| = 0, and
supi(|R−Rm|+ |Ri−Ri,m|) → 0 as m → ∞ (recall (7.13)). Therefore, we have thus
proved (7.36).
Proof of Lemma VII.16. We only prove (7.27). The proof of (7.28) is similar. By
Proposition VI.20, there exists a constant C, such that
(7.37)
Sn(Z − ζ)2
n≤ C
k∈1,nd
E(Zn,k − ζn,k
| F1)2
d
τ=1 k1/2τ
,
where F1 = σ(k : k ∈ Zd, k 1). By the definition of ζ
n,i, (7.37) equals (up to the
multiplicative constant C),
k∈1,nd\1,mnd
E(Zn,k | F1)2
d
τ=1 k1/2τ
+
k∈1,mnd
E(Zn,k − ζn,k
| F1)2
d
τ=1 k1/2τ
≤Zn,0 − ζ
n,0
2+
k∈1,nd
E(Zn,k | F1)2
d
τ=1 k1/2τ
+
k∈1,mnd
E(ζn,k
| F1)2
d
τ=1 k1/2τ
≤ CZn,0 − ζ
n,0
2+ b1/2
n
k∈1,nd
Ak−1d
τ=1 k1/2τ
,
where the last inequality follows from Lemma VII.18 below.
Lemma VII.18. Suppose that in addition to Condition VII.1, E(|0|α) < ∞ for
some α ≥ 2. For all k ∈ Nd, k = 1,
131
E(Zn,k | F1)α
≤ Cb1/2n
Ak−1 ,(7.38)
E(ζn,k
| F1)α
≤ Cb1/2n
Ak−1 .(7.39)
Proof of Lemma VII.18. First, we controlE(Zn,k | F1)
α. For each k ∈ Z
d, intro-
duce the notation
(7.40) Γ(k) := i ∈ Zd : i k ,
and write
Xk =
i∈Γ(k)
ak−ii =
i∈Γ(1)
+
i∈Γ(k)\Γ(1)
ak−ii =: Dk + Tk .
For the sake of simplicity, write D ≡ Dk, T ≡ Tk, and, given a random variable Y , let
EY (·) ≡ E(· | Y ) denote the conditional expectation given the σ-algebra generated
by Y . Since k 1, k = 1, Tk is a non-degenerate random variable. Then,
E(Zn,k | F1) =1
√bn
EDK
x−D − T
bn
− EK
x−D − T
bn
.
Let D be a copy of D, independent of D and T . Then, the above identity becomes,
letting pT denote the density of T ,
1√bnEDED, D
Kx−D − T
bn
−K
x− D − T
bn
= b1/2n
ED
K(t)
pT (x− bnt−D)− pT (x− bnt− D)
dt .
Since pT is Lipschitz, the absolute value of the above term is bounded by
C|K(s)|dsb1/2n ED|D − D|, almost surely. (Here pT depends on k, n, but one
can show that the Lipschitz constant can be chosen independently from k, n. See
e.g. [117], Lemma 1.) To sum up, we have
E(Zn,k | F1)α≤ Cb1/2
n
ED|D − D|
α
≤ Cb1/2n
Dα≤ Cb1/2
nAk−1,
132
where the last inequality follows from (7.13). We have thus proved (7.38). To
prove (7.39), a similar argument yieldsE(ζ
n,k| F1)
α≤ Cb1/2n Ak,mn
with Ak,mn=
(
i∈0,mn−1d,ik−1 a2i)1/2 ≤ Ak−1.
Proof of Lemma VII.17. For random variables Zn,0, Zn,0, ζn,0, ζn,0, we replace the
index ‘n,0’ by ‘n’ for the sake of simplicity. First observe that
(EZn)2 + (Eζn)
2≤ C(p2bn + p2
mnbn) ≤ Cbn ,
where the last step we applied (7.7). Then,
|E(ζ2n− Z
2n)| ≤
K2(y)[pmn(x− bny)− p(x− bny)]dy
+ Cbn
≤ supy
|pmn(y)− p(y)|
K2(s)ds+ Cbn .
≤ C(Bmn+ bn) ,(7.41)
where the last inequality follows from (7.32). Next, write
(7.42)ζ
n− Zn
2
2= EZ
2n− Eζ
2n+ 2(Eζ
2n− E(Znζn)) .
For the last term on the right-hand side of (7.42), observe that E(Znζn) = E(Znζn)−
EZnEζn = E(Znζn) + O(pmn
bn). We claim that E(Znζn) is very close to Eζ2n, under
our restriction on the choice of mn. Indeed,
(7.43) |E(Znζn)− Eζ2n| ≡
E(Znζn)−
K2(y)pmn
(x− bny)dy ,
and,
E(Znζn) =
1
bnKx− y − z
bn
Kx− y
bn
pmn
(y)pmn(z)dydz
=
K(y)EK
y −
X0,mn
bn
pmn
(x− bny)dy .
133
Therefore, (7.43) can be bounded by, since K is Lipschitz,
|K(y)|E
Ky −
X0,mn
bn
−K(y)
pmn(x− bny)dy
≤E| X0,mn
|
bn
|K(y)|pmn
(x− bny)dy ,
and E| X0,mn| ≤ CBmn
by (7.13). To sum up, we have thus shown that (recall that
bn ↓ 0, whence Bmnis dominated by Bmn
/bn), under (7.6),
ζn− Zn
2
2≤ C
Bmn
bn+ bn
.
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134
135
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ABSTRACT
Topics on Max-stable Processes and the Central Limit Theorem
by
Yizao Wang
Chair: Stilian. A. Stoev
This dissertation consists of results in two distinct areas of probability theory.
One is the extreme value theory, the other is the central limit theorem.
In the extreme value theory, the focus is on max-stable processes. Such processes
play an increasingly important role in characterizing and modeling extremal phe-
nomena in finance, environmental sciences and statistical mechanics. In particular,
the association of sum- and max-stable processes and the decomposability of sum-
and max-stable processes are investigated. Besides, the conditional distributions of
max-stable processes are also studied, and a computationally efficient algorithm is
developed. This algorithm has many potential applications in prediction of extremal
phenomena.
In the central limit theorem, the asymptotic normality for partial sums of sta-
tionary random fields is studied, with a focus on the projective conditions on the
dependence. Such conditions, easy to check for many stochastic processes and ran-
1
dom fields, have recently drawn many attentions for (one-dimensional) time series
models in statistics and econometrics. Here, the focus is on (high-dimensional) sta-
tionary random fields. In particular, a general central limit theorem for stationary
random fields and orthomartingales is established. The method is then extended to
establish the asymptotic normality for the kernel density estimator of linear random