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INSTITUTE OF PHYSICS PUBLISHING NONLINEARITY

Nonlinearity 18 (2005) 1955–1965 doi:10.1088/0951-7715/18/5/005

Central limit theorems for contractive Markov chains

Andreas Nordvall Lagerås and Orjan Stenflo

Department of Mathematics, Stockholm University, SE-10691 Stockholm, Sweden

E-mail: [email protected] and [email protected]

Received 9 February 2005, in final form 3 May 2005Published 6 June 2005Online at stacks.iop.org/Non/18/1955

Recommended by C P Dettmann

AbstractWe prove limit theorems for Markov chains under (local) contraction conditions.As a corollary we obtain a central limit theorem for Markov chains associatedwith iterated function systems with contractive maps and place-dependentDini-continuous probabilities.

Mathematics Subject Classification: 60F05, 60J05, 60B10, 37H99

1. Introduction

Let (X, d) be a compact metric space, typically a closed and bounded subset of R or R2

with the Euclidean metric and let {wi}Ni=1 be a family of (strict) contraction maps on X,i.e. there exists a constant c < 1 such that d(wi(x), wi(y)) � cd(x, y), for any x, y ∈ X

and integer 1 � i � N . Such a system is called an iterated function system (IFS) (see [1]).Hutchinson [12] and Barnsley and Demko [1] introduced these objects in order to describefractals. It is easy to see that there exists a unique compact set K that is invariant for the IFSin the sense that K = ⋃N

i=1 wi(K). The set K is called the fractal set, or attractor, associatedwith the IFS. If the maps wi are non-degenerate and affine and the sets wi(K), 1 � i � N , are‘essentially’ disjoint, then K will have the characteristic ‘self-similar’ property of a fractal.The huge class of examples of fractals that can be described in this way includes the Sierpinskigasket, Barnsley’s fern, the Cantor set and many, many others. Despite fractals being totallydeterministic objects, the simplest way of drawing pictures of fractals is often via Barnsley’s‘random iteration algorithm’: attach probabilities, pi , to each map wi (

∑i pi = 1). Choose

a starting point Z0(x) := x ∈ X. Choose a function, wI1 , at random from the IFS, withP(wI1 = wk) = pk . Let Z1(x) = wI1(x). Next, independently, choose a function, wI2 , inthe same manner and let Z2(x) = wI2(Z1(x)) = wI2 ◦ wI1(x). Repeat this ‘random iteration’procedure inductively and define Zn(x) = wIn

◦ wIn−1 ◦ · · · ◦ wI1(x). The random sequence{Zn(x)} forms a Markov chain with a unique stationary probability distribution, µ, supportedon K . Since ∑n−1

k=0 f (Zk(x))

n→

∫f dµ a.s.,

0951-7715/05/051955+11$30.00 © 2005 IOP Publishing Ltd and London Mathematical Society Printed in the UK 1955

http://dx.doi.org/10.1088/0951-7715/18/5/005

mailto: [email protected]

mailto: [email protected]

http://stacks.iop.org/no/18/1955

1956 A N Lagerås and Orjan Stenflo

as n → ∞, for any real-valued continuous function f on X, by Birkhoff’s ergodic theorem(note that x can be chosen to be any fixed point by the contraction assumption), we will ‘drawa picture of the attractor K’ by ‘plotting’ the orbit {Zn(x)}, possibly ignoring some of thefirst points in order to reach the stationary regime. This algorithm will be an efficient wayof ‘drawing a picture of K’ provided the probabilities are chosen in such a way as to makethe stationary distribution as uniform as possible on K and the stationary state is reachedsufficiently fast. The choice of pi can sometimes be made by inspection, by searching for astationary distribution with the same dimension as K itself. The convergence rates towardsthe stationary state are ‘heuristically justified’ by central limit theorems (CLTs), where

1√n

n−1∑k=0

(f (Zk(x)) −

∫f dµ

)

converges in distribution to the normal distribution for f belonging to some suitably rich classof real-valued functions on X, or by stronger forms of CLTs, the so-called invariance principlesor functional CLTs, where the stochastic process

1√n

[nt]−1∑k=0

(f (Zk(x)) −

∫f dµ

), 0 � t � 1 (1)

converges in distribution to a Brownian motion. (Here [x] denotes the integer part of x.) Notethat expression (1) above is a function-valued random element. See [5] for details about theconcept of convergence in distribution for function-valued random elements.

The purpose of this paper is to study Markov chains generated by IFSs with place-dependent probabilities. (Such Markov chains have also been studied under the name ‘randomsystems with complete connections’, see [13].) We are given a set of contraction maps {wi},with associated continuous functions pi = pi(x), where pi : X → (0, 1), with

∑i pi(x) = 1,

for any x ∈ X. The Markov chains are characterized by the transfer operator T defined forreal-valued measurable functions f on X by Tf (x) = ∑

i pi(x)f (wi(x)). Intuitively, theMarkov chains considered are generated by fixing a starting point x and letting Z0(x) := x,and inductively letting Zn+1(x) := wi(Zn(x)) with probability pi(Zn(x)) for n � 0.

One motivation for studying such chains is that it gives more freedom when trying togenerate a ‘uniform’ stationary probability distribution on K . Such Markov chains also arisenaturally in the thermodynamic formalism of statistical mechanics. It is well known that theydo not necessarily possess a unique stationary distribution (see [4,6,20,26,27]), but with someadditional regularity conditions, uniqueness holds (see [11, 14, 27, 28]).

The operator T (without the normalizing condition∑

i pi(x) = 1) is known as theRuelle–Perron–Frobenius operator. Fan and Lau [10] proved a limit theorem for iteratesof the Ruelle–Perron–Frobenius operator under the Dini-continuity assumptions on the pi ,by lifting a similar result from Walters [29] on symbolic spaces. (Recall that pi is Dini-continuous if

∫ 10 (�pi

(t)/t) dt < ∞, or equivalently∑∞

n=0 �pi(cn) < ∞, for some (and thus

all) 0 < c < 1, where �pi(t) := supd(x,y)�t |pi(x) − pi(y)| is the modulus of uniform

continuity of pi). Uniqueness in stationary distributions still holds (in the normalized cases)if the contraction assumptions of the wi are relaxed to ‘average contraction’ under the Dini-continuity assumption (see [2,17]) but information about rates of convergence in these ‘averagecontractive’ cases seems to be unknown.

The Dini-condition is somewhat stronger than the weakest known conditions foruniqueness in stationary probability distributions (in the normalized cases with strictcontraction maps), but weaker than, e.g., Holder-continuity.

CLTs for contractive Markov chains 1957

In the Dini-continuous cases it follows that the unique equilibrium measure will have theGibbs (approximation) property (see [10]). This property is of importance when analysing themultidimensional spectra of measures.

In this paper we will prove the perhaps initially surprising fact (corollary 2) thatMarkov chains generated by IFSs with Dini-continuous probabilities obey a CLT, despitethe well-known fact that such Markov chains do not typically converge with an exponentialrate. Our main result, theorem 1, expresses this in a natural generality.

CLTs/functional CLTs for iterated random functions under conditions that implyexponential (or other rapid) rates of convergence have previously been proved in,e.g., [3, 15, 16, 22, 30, 31]. We discuss the connection between some of these results and ourresults in remarks 4 and 6.

2. Preliminaries

Let B denote the Borel σ -field generated by the metric d, and let P : X × B → [0, 1]be a transition probability. That is, for each x ∈ X, P(x, ·) is a probability measure on (X, B)

and for each A ∈ B, P(·, A) is B-measurable. The transition probability generates a Markovchain with transfer operator defined by Tf (x) = ∫

Xf (y)P(x, dy) for real-valued measurable

functions f on X. A probability measure µ is stationary for P if µ(·) = ∫X

P(x, ·) dµ(x).There are several ways of representing a Markov chain with a given transfer operator.

One common way is to find a measurable function w : X × [0, 1] → X, let {Ij }∞j=1 be asequence of independent random variables uniformly distributed in [0, 1], and consider therandom dynamical system defined by

Zn(x) := wIn◦ wIn−1 ◦ · · · ◦ wI1(x), n � 1, Z0(x) := x,

for any x ∈ X, where

ws(x) = w(x, s).

It is always possible to find such a representation, w, such that the transition probabilitygenerated by {Zn} is P, i.e. Ef (Zn(x)) = T nf (x), for any x, n and f (see [19]).

For two fixed points x, y ∈ X and x = (x, y) we can consider the Markov chain {Zn(x)},on X2, where Zn(x) := (Zn(x), Zn(y)). When proving theorems based on contractionconditions we are typically interested in representations that minimize d(Zn(x), Zn(y))

(in some average sense).More generally, if W : X2 × [0, 1] → X2, is a measurable map and {Ij }∞j=1 is a sequence

of independent random variables uniformly distributed in [0, 1], we will consider the randomdynamical system defined by

Zn(x) := WIn◦ WIn−1 ◦ · · · ◦ WI1(x), n � 1, Z0(x) := x, (2)

where Ws(x) = W(x, s), such that, for any x = (x, y) ∈ X2, the Markov chainZn(x) =: (Z

(x,y)n (x), Z

(x,y)n (y)) on X2 has marginals Pn(x, ·) = P(Z

(x,y)n (x) ∈ ·), and

Pn(y, ·) = P(Z(x,y)n (y) ∈ ·), for any n.

Thus {Z(x,y)n (x)} and {Z(x,y)

n (y)} denote two Markov chains on X, defined on the sameprobability space, with the former starting at x ∈ X and the latter starting at y ∈ X, both withtransition probability P.

Let dw be the Monge–Kantorovich metric defined by dw(π, ν) = sup(∫

f d(π − ν);f : X → R, |f (x) − f (y)| � d(x, y) ∀x, y), for probability measures π and ν on X. TheMonge–Kantorovich metric metrizes the topology of weak convergence on the set of probability


measures on X (see [9]). It follows from the definitions that for any stationary probabilitymeasure µ, we have

dw(Pn(x, ·), µ(·)) � supx,y∈X

Ed(Z(x,y)n (x), Z(x,y)

n (y)). (3)

Therefore if supx,y Ed(Z(x,y)n (x), Z

(x,y)n (y)) → 0 as n → ∞, then there is a unique stationary

distribution for P.We will sometimes drop the upper index, i.e. write Zn(x) instead of Z

(x,y)n (x) etc, when

we are not interested in the joint distribution of the pair (Z(x,y)n (x), Z

(x,y)n (y)).

The following proposition gives sufficient conditions for the existence of a CLT.

Proposition 1. Suppose there exists a unique stationary distribution µ for P, and let f bea real-valued measurable function on X with ‖f ‖2

L2 = ∫f 2 dµ < ∞. Suppose that for

some δ > 0,

limn→∞ n−1/2(log n)1+δ sup

x,y∈X

E

n−1∑k=0

| f (Z(x,y)

k (x)) − f (Z(x,y)

k (y)) | = 0. (4)

Let

Sxn =

n−1∑k=0

(f (Zk(x)) − Ef (Zk(x))),

Sx,µn =

n−1∑k=0

(f (Zk(x)) −

∫f dµ

)

and

Bxn (t) = Sx

[nt]√n

, 0 � t � 1,

Bx,µn (t) = S

x,µ[nt]√n

, 0 � t � 1.

Then the limit

σ 2 = σ 2(f ) := limn→∞

1

nE[(SZ

n )2] (5)

exists and is finite, where Z is a µ-distributed random variable, independent of {Ij }∞j=1.Furthermore, if B = {B(t) : 0 � t � 1} denotes the standard Brownian motion, then

Bxn

d→ σB (6)

and

Bx,µn

d→ σB, (7)

as n → ∞, for any x ∈ X, whered→ denotes convergence in distribution for random elements

taking values in the space of right-continuous functions on [0, 1] with left-hand limits equippedwith the Skorokhod topology.

Remark 1. Proposition 1 above is valid when (X, B) is a general measurable space.

Remark 2. General CLTs for Markov chains started at a point have been proved by Derriennicand Lin [7]. Proposition 1 complements their result in cases of ‘uniform’ ergodicity. The proofof proposition 1, given later, relies on a slightly stronger result by Peligrad and Utev [23] for


Markov chains starting according to the unique stationary probability distribution. Theoremsabout convergence, allowing Markov chains to start at a point, are important in the theory forMarkov chain–Monte Carlo methods.

Remark 3. In an earlier draft of this paper we proved a weaker (non-functional) form of theCLT in proposition 1, where our result was based on a CLT by Maxwell and Woodroofe [21].The recent paper by Peligrad and Utev [23], which was helpfully pointed out to us by a referee,enabled us to state our CLT in the current functional CLT form.

Remark 4. Wu and Woodroofe considered general state spaces in [31]. The conditions intheir CLT (theorem 2) imply (4) in the case of a compact X. This can be seen as follows: theirproof of this theorem amounts to showing that

∑∞n=0 ‖T nf ‖L2 < ∞, for centred functions f .

Restricting X to be compact allows a strengthening of their lemma 3, so that its result holdseven when starting {Zk(x)} from a point. With some minor modifications to the proof, it ispossible to show that

∑∞n=0 supx,y E|f (Z

(x,y)n (x)) − f (Z

(x,y)n (y))| < ∞. Thus the conditions

of our proposition 1 hold.

Checking the L2 boundedness condition could be difficult if we have no a priori informa-tion about the (possibly non-unique) stationary measures. The following corollary circumventsthese problems and might therefore be more directly applicable in our case when (X, d) iscompact.

Corollary 1. If

limn→∞ sup

x,y∈X


n (y)) = 0, (8)

then there exists a unique stationary distribution µ for P.Let f be a real-valued continuous function on X. Suppose �f : R

+ → R+ is an

increasing concave function with �f (t) � supd(x,y)�t |f (x) − f (y)|, for any t � 0 andsuppose, in addition to (8), that for some δ > 0,

limn→∞

√n(log n)1+δ�f

(sup

x,y∈X


n (y))

)= 0 (9)

also holds, then the conclusions of proposition 1 hold for f , i.e. the limit (5) exists for f andis finite and (6) and (7) hold.

Remark 5. The function �f may thus be chosen to be the modulus of uniform continuity off in cases when this function is concave.

Remark 6. If supx,y∈X Ed(Z(x,y)n (x), Z

(x,y)n (y)) ∼ O(cn), for some constant c < 1,

satisfied for instance the average-contractive IFSs with place-independent probabilities, thenit follows from corollary 1 that the CLT holds with respect to any f of modulus ofuniform continuity �f , of order �f (cn) ∼ o(1/

√n(log n)1+δ). This condition is satisfied

by, e.g., Dini-continuous functions f . Corollary 1 thus strengthens theorem 2.4. of [3](who considered Lipschitz-continuous f ). Wu and Shao [30] considered functions f

that are stochastically Dini-continuous with respect to the stationary distribution. (Itshould be noted that [3] and [30] treated average contractive IFSs on more general metricspaces.)

Proof (proposition 1). Let f ∈ L2(µ) be a real-valued measurable function on X

satisfying assumption (4).


Since

∞∑n=1

1

n3/2

∥∥∥∥∥n−1∑k=0

T k

(f −

∫f dµ

) ∥∥∥∥∥L2

=∞∑

n=1

1

n3/2

∥∥∥∥∥n−1∑k=0

(T kf −

∫f dµ

) ∥∥∥∥∥L2

�∞∑

n=1

1

n3/2supx∈X

∣∣∣∣∣n−1∑k=0

(T kf (x) −

∫f dµ

) ∣∣∣∣∣=

∞∑n=1

1

n3/2supx∈X

∣∣∣∣∣n−1∑k=0

(T kf (x) −

∫T kf dµ

) ∣∣∣∣∣�

∞∑n=1

1

n3/2sup

x,y∈X

∣∣∣∣∣n−1∑k=0

(T kf (x) − T kf (y)

) ∣∣∣∣∣=

∞∑n=1

1

n3/2sup

x,y∈X

∣∣∣∣∣En−1∑k=0

(f (Z(x,y)

k (x)) − f (Z(x,y)

k (y)))

∣∣∣∣∣�

∞∑n=1

1

n3/2sup

x,y∈X

E

n−1∑k=0

∣∣∣∣∣f (Z(x,y)

k (x)) − f (Z(x,y)

k (y))

∣∣∣∣∣ < ∞,

it follows from theorem 1.1 of [23] that σ 2 = limn→∞(1/n)E[(SZn )2] exists and is finite,

and BZ,µn

d→ σB, where Z is a µ-distributed random variable, independent of {Ij }∞j=1.By Chebyshev’s inequality,

P

(sup

0�t�1|Bx,µ

n (t) − BZ,µn (t)| � ε

)

= P

(max

0�m�n

1√n

∣∣∣∣∣m−1∑k=0

(f (Z(x,Z)k (x)) − f (Z

(x,Z)k (Z)))

∣∣∣∣∣ � ε

)

� P

(1√n

max0�m�n

m−1∑k=0

∣∣∣∣∣f (Z(x,Z)k (x)) − f (Z

(x,Z)k (Z))

∣∣∣∣∣ � ε

)

� P

(1√n

n−1∑k=0

∣∣∣∣∣f (Z(x,Z)k (x)) − f (Z

(x,Z)k (Z))

∣∣∣∣∣ � ε

)

� 1

ε√

nE

n−1∑k=0

∣∣∣∣∣f (Z(x,Z)k (x)) − f (Z

(x,Z)k (Z))

∣∣∣∣∣� 1

ε√

nsup

x,y∈X

E

n−1∑k=0

∣∣∣∣∣f (Z(x,y)

k (x)) − f (Z(x,y)

k (y))

∣∣∣∣∣ → 0,

as n → ∞. By theorem 4.1 in [5], Bx,µn

d→ σB.


The difference between Sxn and S

x,µn lies in how the summands are centred. The difference

is negligible in the limit:

sup0�t�1

|Bx,µn (t) − Bx

n (t)| = max0�m�n

1√n

∣∣∣∣∣m−1∑k=0

(Ef (Zk(x)) −

∫f dµ

) ∣∣∣∣∣� 1√

nmax

0�m�nE

∣∣∣∣∣m−1∑k=0

(f (Zk(x)) −

∫f (Zk(y)) dµ(y)

) ∣∣∣∣∣� 1√

nE

n−1∑k=0

∣∣∣∣∣f (Zk(x)) −∫

f (Zk(y)) dµ(y)

∣∣∣∣∣� 1√

nsup

x,y∈X

E

n−1∑k=0

∣∣∣∣∣f (Z(x,y)

k (x)) − f (Z(x,y)

k (y))

∣∣∣∣∣ → 0,

as n → ∞. Thus also Bxn

d→ σB. �

Proof (corollary 1). The first part of the corollary follows from (3) above.For the proof of the second part of corollary 1, first note that by assumption (9),

�f

(sup

x,y∈X


n (y))

)∼ o

(1√

n(log n)1+δ

),

implying that

n−1∑k=0

�f

(sup

x,y∈X

Ed(Z(x,y)

k (x), Z(x,y)

k (y))

)∼ o

( √n

(log n)1+δ

).

(To see this, note that the derivative F ′(t) of F(t) = √t/(log t)1+δ satisfies F ′(t) �

1/(3√

t(log t)1+δ), for large t .)Thus,

limn→∞ n−1/2(log n)1+δ

n−1∑k=0

�f

(sup

x,y∈X

Ed(Z(x,y)

k (x), Z(x,y)

k (y))

)= 0.

Since by the definition of �f and Jensen’s inequality,

�f

(sup

x,y∈X


n (y))

)� sup

x,y∈X

�f (Ed(Z(x,y)n (x), Z(x,y)

n (y)))

� supx,y∈X

E�f (d(Z(x,y)n (x), Z(x,y)

n (y)))

� supx,y∈X

E|f (Z(x,y)n (x)) − f (Z(x,y)

n (y))|

and

n−1∑k=0

supx,y∈X

E|f (Z(x,y)

k (x)) − f (Z(x,y)

k (y))| �supx,y∈X

E

n−1∑k=0

|f (Z(x,y)

k (x)) − f (Z(x,y)

k (y))|,


we see that an application of proposition 1 completes the proof of the second part ofcorollary 1. �

3. Main results

Theorem 1. Let W : X2 × [0, 1] → X2 be a measurable map such that for any fixed(x, y) ∈ X2 the map W(x, y, ·) := (W(x,y)(x), W(x,y)(y))(·) defines random variables withP(W(x,y)(x) ∈ ·) = P(x, ·) and P(W(x,y)(y) ∈ ·) = P(y, ·), where P denotes the uniformprobability measure on the Borel subsets of [0, 1].

Let � : [0, ∞) → [0, 1), be an increasing function with �(0) = 0. Suppose there existsa constant c < 1, such that

P(d(W(x,y)(x), W(x,y)(y)) � cd(x, y)) � 1 − �(d(x, y)), (10)

for any two points x, y ∈ X.Then

(i) (Distributional stability theorem)

dw(Pn(x, ·), µ(·)) � supx,y∈X


n (y)) � EDn, (11)

for any stationary probability distribution µ, where Dn is a homogeneous Markov chain withD0 = diam(X) := supx,y d(x, y),

P(Dn+1 = ct | Dn = t) = 1 − �(t)

andP(Dn+1 = diam(X) | Dn = t) = �(t),

for any 0 � t � diam(X).If

∞∑n=1

n∏k=1

(1 − �(ck)) = ∞, (12)

then EDn → 0 and thus by corollary 1 there is a unique stationary distribution, µ.(ii) (Central limit theorem)

If∑∞

k=0 �(ck) < ∞, then the conclusions of proposition 1 hold for any Holder-continuousfunction f with exponent α > 1

2 .

Proof (theorem 1(i)). Fix two points x and y in X. Define Z(x,y)

0 (x) = x, Z(x,y)

0 (y) = y andinductively

Z(x,y)n (x) = W(Z

(x,y)

n−1 (x),Z(x,y)

n−1 (y))(Z(x,y)

n−1 (x))

andZ(x,y)

n (y) = W(Z(x,y)

n−1 (x),Z(x,y)

n−1 (y))(Z(x,y)

n−1 (y)),

as in (2). Then Z(x,y)n (x) and Z

(x,y)n (y) are random variables such that Ef (Z

(x,y)n (x)) = T nf (x)

and Ef (Z(x,y)n (y)) = T nf (y), for any n.

We have from assumption (10) that

P(d(Z(x,y)n (x), Z(x,y)

n (y)) � ct | d(Z(x,y)

n−1 (x), Z(x,y)

n−1 (y)) � t)

� 1 − �(t) = P(Dn = ct |Dn−1 = t),


for any t ∈ {ckdiam(X)}∞k=0. (Note that Dn takes values in the discrete state space{ckdiam(X)}∞k=0.)

Dn is therefore stochastically larger thand(Z(x,y)n (x), Z

(x,y)n (y)), and consequentlyEDn �

Ed(Z(x,y)n (x), Z

(x,y)n (y)), for any x, y ∈ X. The other inequality of (11) follows from (3).

Since {Dn} is a non-ergodic Markov chain under condition (12) (see [24], p 80), it followsthat EDn → 0 as n → ∞, if (12) holds, and we have thus proved theorem 1(i).

In order to prove theorem 1(ii), we first observe that it is well known that∑∞

k=0 �(ck) < ∞implies that Dn is transient (see [24], p 80). Therefore (see [25], p 575),

∑∞k=0 P(Dk =

diam(X)) < ∞ and it follows that

∞∑k=0

EDk =∞∑

k=0

k∑j=0

cj diam(X)P (Dk = cj diam(X))

� diam(X)

∞∑k=0

k∑j=0

cjP (Dk−j = diam(X))

= diam(X)

1 − c

∞∑k=0

P(Dk = diam(X)) < ∞.

By stochastic monotonicity EDk is decreasing, and thus∑n

k=1 EDk � nEDn, for any n. Thisimplies that EDn � c0/n, for c0 := ∑∞

k=0 EDk .Thus supx,y Ed(Z

(x,y)n (x), Z

(x,y)n (y)) � c0/n, for any n � 1. If f is a Holder-continuous

function on X, with modulus of uniform continuity �f satisfying �f (t) � c1tα , for some

constants c1 and α > 12 , and any t � 0, it follows that for any δ > 0,

limn→∞

√n(log n)1+δ�f

(sup

x,y∈X


n (y))

)

� limn→∞

√n(log n)1+δc1

(c0

n

)α

= 0.

An application of corollary 1 now completes the proof of theorem 1(ii). �

4. IFSs with place-dependent probabilities

Let {wi}∞i=1 be a set of strictly contracting maps, i.e. there exist a constant c < 1 such thatd(wi(x), wi(y) � cd(x, y), for any x, y ∈ X and any integer i. Let {pi(x)}∞i=1 be associatedplace-dependent probabilities, i.e. non-negative continuous functions, with

∑i pi(x) = 1,

for any x ∈ X. This system defines a Markov chain with transfer operator defined byTf (x) = ∑∞

i=1 pi(x)f (wi(x)), for real-valued measurable functions f on X.Let

�(t) = 1

2sup

d(x,y)�t

∞∑i=1

|pi(x) − pi(y)| = 1 − infd(x,y)�t

∞∑i=1

min(pi(x), pi(y)) (13)

and let for any two points x, y ∈ X, W(x,y)(x) and W(x,y)(y) be random variables defined by

P(W(x,y)(x) = wi(x), W(x,y)(y) = wi(y)) = min(pi(x), pi(y)) (14)


and

P(W(x,y)(x) = wi(x), W(x,y)(y) = wj(y))

= (pi(x) − min(pi(x), pi(y)))(pj (y) − min(pj (x), pj (y)))

1 − ∑∞k=1 min(pk(x), pk(y))

, (15)

when i = j . (If pi(x) = pi(y), ∀ i, then we understand the expression in (15) as zero.)It is straightforward to check that by construction P(W(x,y)(x) = wi(x)) = pi(x), and

P(W(x,y)(y) = wj(y)

) = pj (y) for any i and j .It follows from (14) that

P(d(W(x,y)(x), W(x,y)(y)) � cd(x, y)) �∞∑i=1

min(pi(x), pi(y)) � 1 − �(d(x, y)),

and we may therefore apply theorem 1 to obtain the following.

Corollary 2. Let {wi}∞i=1 be an IFS with strictly contractive maps, and let {pi(x)} be associatedplace-dependent probabilities. Then the conclusions of theorem 1 hold with � defined asin (13) above.

Let us illustrate the above corollary with an example.

Example 1. Let w1 and w2 be two maps from [0, 1] into itself defined by

w1(x) = βx and w2(x) = βx + (1 − β),

where 0 < β < 1 is a constant parameter. Consider the Markov chain with transfer operatorT : C([0, 1]) → C([0, 1]) defined by

Tf (x) = p(x)f (w1(x)) + (1 − p(x))f (w2(x)), f ∈ C([0, 1]),

where p : [0, 1] → (0, 1), is a continuous function with modulus of uniform continuity� = �p.

The case when p(x) ≡ 12 and β = 1

2 , where the uniform distribution on [0, 1] is the uniquestationary distribution, and the case when p(x) ≡ 1

2 and β = 13 , where the uniform distribution

on the (middle third) Cantor set is the unique stationary distribution, are two important particularcases of this model.

For general p, Markov chains of this form always possess a stationary probabilitydistribution, but they may possess more than one stationary probability distribution (see [26]).

From theorem 1 it follows that the distribution will be unique (for any fixed value of theparameter β) provided (12) holds, and this theorem also enables us to quantify the rate ofconvergence as a function of the modulus of uniform continuity of p. It also follows that thisMarkov chain will obey the functional CLT (6) and (7) for Holder-continuous functions f

with exponent α > 12 provided p is Dini-continuous. Observe that our conditions are only

sufficient. It is an interesting open problem to try to find critical smoothness properties of p

to ensure a unique stationary measure and a CLT.

Remark 7. If X = {1, . . . , N}N and for two elements x = x0x1 . . . and y = y0y1 . . . inX, we define d(x, y) := 2− min(k�0; xk =yk) if x = y, and d(x, y) := 0 if x = y, then(X, d) is a compact metric space. Let g be a continuous function from X to (0, 1], suchthat

∑Nx0=1 g(x0x1 . . .) = 1 for all x1x2 . . . ∈ X. g describes an IFS with place-dependent

probabilities: {(X, d), wi(x), pi(x), i ∈ {1, . . . , N}}, where wi(x) = ix and pi(x) = g(ix),and corollary 2 applies. This generalizes theorem 1 in [28] and also implies a CLT for theassociated Markov chains under the ‘summable variations’ condition used in [8] or [29].Stationary probability measures for such Markov chains are sometimes called g-measures, aconcept coined by Keane [18].


Acknowledgments

We are grateful to the referees for pointing out useful references and for providing us withcomments for improving this paper.

References

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Barnsley M F, Demko S G, Elton J H and Geronimo J S 1989 Ann. Inst. H Poincare Probab. Stat. 25 589–90(erratum)

[3] Benda M 1998 A central limit theorem for contractive stochastic dynamical systems J. Appl. Probab. 35 200–5[4] Berger N, Hoffman C and Sidoravicius V 2003 Nonuniqueness for specifications in 2+ε Preprint

math PR/0312344[5] Billingsley P 1968 Convergence of Probability Measures (New York: Wiley)[6] Bramson M and Kalikow S 1993 Non-uniqueness in g-functions Israel J. Math. 84 153–60[7] Derriennic Y and Lin M 2003 The central limit theorem for Markov chains started at a point Probab. Theory

Rel. Fields 125 73–6[8] Doeblin W and Fortet R 1937 Sur les chaines a liaisons completes Bull. Soc. Math. France 65 132–48[9] Dudley R M 1989 Real Analysis and Probability (Pacific Grove, CA: Wadsworth and Brooks/Cole)

[10] Fan A H and Lau K-S 1999 Iterated function system and Ruelle operator J. Math. Anal. Appl. 231 319–44[11] Harris T E 1955 On chains of infinite order Pac. J. Math. 5 707–24[12] Hutchinson J E 1981 Fractals and self-similarity Indiana Univ. Math. J. 30 713–47[13] Iosifescu M and Grigorescu S 1990 Dependence with Complete Connections and its Applications (Cambridge:

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random matrices PhD Thesis Institut Mittag-Leffler, Djursholm, Sweden[16] Kaijser T 1979 Another central limit theorem for random systems with complete connections Rev. Roum. Math.

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http://arxiv.org/abs/math~PR/0312344

Applied Probability Trust (15 September 2005)

A RENEWAL PROCESS TYPE EXPRESSION FOR THE MO-

MENTS OF INVERSE SUBORDINATORS

ANDREAS NORDVALL LAGERAS,∗ Stockholm University

Abstract

We define an inverse subordinator as the passage times of a subordinator to

increasing levels. It has previously been noted that such processes have many

similarities with renewal processes. Here we present an expression for the joint

moments of the increments of an inverse subordinator. This is an analogue of

a result for renewal processes. The main tool is a theorem on the processes

which are both renewal processes and Cox processes.

Keywords: Subordinator; Passage time; Renewal theory; Cox process; Local

time

AMS 2000 Subject Classification: Primary 60K05

Secondary 60G51; 60G55; 60E07

1. Introduction

Subordinators are non-decreasing processes with independent and stationary incre-

ments. The corresponding processes in discrete time are the partial-sum processes

with positive, independent and identically distributed summands. Renewal processes

can be considered to be passage times of partial-sum processes to increasing levels.

Analogously we can define a process by the passage times of a subordinator. We call

such a process an inverse subordinator.

The inverse subordinators appear in diverse areas of probability theory: As Bertoin

[2] notes, the local times of a large class of well-behaved Markov processes are really

inverse subordinators, and any inverse subordinator is the local time of some Markov

process. It is well known, see Karatzas and Shreve [9], that the local time of the

Brownian motion is the inverse of a 1/2-stable subordinator. Inverses of α-stable

∗ Postal address: Department of Mathematics, Stockholm University, SE-10691 Stockholm, Sweden

Email address: [email protected]

1

2 ANDREAS NORDVALL LAGERAS

subordinators with 0 < α < 1 arise as limiting processes of occupation times of

Markov processes, see Bingham [4]. Some recent applications of inverse subordinators

in stochastic models can be found in [8], [11] and [14]. Kaj and Martin-Lof [8] consider

superposition and scaling of inverse subordinators with applications in queueing theory,

Kozlova and Salminen [11] uses diffusion local time as input in a so-called storage

process and Winkel [14] uses inverse subordinators in financial modeling.

In this paper we study some general distributional properties of inverse subordi-

nators, using renewal theory and some theory about Cox processes. In particular we

find an expression for the joint moments of their increments. Other results for inverse

subordinators analogous to those in renewal theory has been proved by Bertoin, van

Harn and Steutel, see [3] and [7].

Some well-known results on subordinators and infinitely divisible distributions on

the positive real line are given in section 2 of this paper. Section 3 introduces the

inverse subordinators and hints that they may have properties similar to the renewal

processes. In section 4 the main result is given: An expression for the joint moments

of the increments of an inverse subordinator. This is proved using a representation

of the class of point processes that are both Cox processes and renewal processes.

With this representation one can also give an alternative proof of the fact that inverse

subordinators can be delayed to be given stationary increments, see [7]. We also provide

a bound of the upper tail of the marginal distribution of an inverse subordinator.

Finally, section 5 exemplifies the results with three types of inverse subordinators.

2. Some basic facts about subordinators

The following results on infinitely divisible distributions and Levy processes can be

found in [13]. Let {Yt} be a Levy process, i.e. a stochastic process in continuous

time with Y0 = 0 and stationary and independent increments. The distribution F of

Y1 is necessarily infinitely divisible, i.e. for all n ∈ N there is a distribution Fn such

that F ?nn = F . Here F ?nn is the n-fold convolution of Fn. The converse is also true:

Given an infinitely divisible distribution F there is a Levy process {Yt} such that the

distribution of Y1 is F . Define F ?t for positive, non-integer t by F ?t(x) = P (Yt ≤ x).

A renewal type expression for the moments of inverse subordinators 3

One recognizes that Fn = F ?1/n.

If one restricts F to be a distribution on R+ then the increments of {Yt} are all

non-negative. Levy processes with non-negative increments are called subordinators.

It is well known that the Laplace-Stieltjes transform of F ?t, where F is an infinitely

divisible distribution on R+, can be written

F ?t(u) =∫ ∞

0

e−uxF ?t(dx) = e−tψ(u) = F (u)t,

where ψ(u) is called the Levy exponent. It can be written in the following form

ψ(u) = δu+∫ ∞

0

(1− e−ux)ν(dx),

where δ ≥ 0 is called the drift and ν(dx) is called the Levy measure. If Y1 has drift δ

then Y1 − δ has drift 0. If∫∞0ν(dx) < ∞ then {Yt} is a compound Poisson process,

with drift if δ > 0, and thus only makes a finite number of jumps in any finite interval.

We call a function π the Levy density if ν(A) =∫Aπ(x)dx. If we define µ = E[Y1],

then µ = δ +∫∞0xν(dx). Since ψ′(u) = δ +

∫∞0e−uxxν(dx), we have

ψ′(0) = µ and ψ′(u) ↘ δ as u↗∞. (1)

Some parts of the reasoning in the following sections do not apply to compound Poisson

processes without drift. Therefore we will henceforth, albeit somewhat artificially,

exclude the compound Poisson processes without drift when referring to subordinators.

3. Inverse subordinators and renewal processes

It is advantageous to recall some results on renewal processes before a more thorough

study of subordinators and their inverses. Let X2, X3, . . . be a sequence of independent

and identically distributed (strictly) positive random variables with distribution F ,

and X1 a positive random variable with distribution H, independent of X2, X3, . . . .

Let S0 = 0 and Sn =∑nk=1Xk, and we call {Sn} a partial-sum process. Given a

partial-sum process we define the renewal process with interarrival distribution F by

Nt = min(n ∈ N : Sn > t) − 1. The −1 in the definition comes from the fact that we

do not want to count the renewal at the origin, as is sometimes done. If F = H then

{Nt} is called an ordinary renewal process.


It is well known that {Nt} has stationary increments if and only if H(x) = 1µ

∫ x0

(1−

F (y))dy, where µ = E[X2] =∫∞0

(1−F (x))dx, and µ necessarily is finite, see [5]. Then

one also has

E[X1] =E[X2

2 ]2µ

, (2)

and the Laplace-Stieltjes transform of H is

H(s) =1µs

(1− F (s)). (3)

We note, as in [3], that subordinators are continuous time analogues of partial-sum

processes. A Levy process sampled at equidistant time points does produce a partial-

sum process with infinitely divisible F , e.g. Yn =∑nk=1(Yk − Yk−1), when the time

points are the integers. As the renewal processes are integer valued inverses to partial-

sum processes, an inverse of a subordinator could be expected to have some properties

similar to renewal processes. Given a subordinator {Yt}, we define τt = inf(τ > 0 :

Yτ > t), and call the process {τt}t≥0 the inverse subordinator.

The properties of the paths of {τt} differ depending on {Yt}. Let us first consider a

compound Poisson process {Yt} with drift δ > 0. Since a jump in {Yt} corresponds to

a flat period in its inverse, {τt} alternates between linear increasing with slope 1δ for

exponential periods of time and being constant for periods of time with lengths drawn

from the compounding distribution, with all these periods having independent lengths.

It is more tricky when {Yt} is not compound Poisson and the drift is zero. Due to the

fact that {Yt} in this case makes an infinite number of jumps in any finite interval, the

trajectories of {τt} are continuous singular almost surely.

Now we will show that {τt} can be arbitrarily closely approximated by a scaled

renewal process. For any c > 0, let {Y ct } be defined by Y ct = Yt/c. Note that {Y ct } is

a subordinator with Y c1 ∼ F ?1/c. Also define the renewal process N ct = min(n ∈ N :


Y cn > t)− 1. Since

cτt = c inf(τ > 0 : Yτ > t)

= inf(cτ > 0 : Yτ > t)

= inf(τ > 0 : Yτ/c > t)

= inf(τ > 0 : Y cτ > t)

≥ min(n ∈ N : Y cn > t)− 1

≥ inf(τ > 0 : Y cτ > t)− 1

= cτt − 1,

τt = 1cN

ct + rt, where 0 ≤ rt ≤ 1

c ,

and the approximation becomes arbitrarily good as c→∞. This result suggests that

the inverse subordinators may have some properties similar to renewal processes. That

this is in fact true will be shown in the following section.

An important function in the theory of renewal processes is the so called renewal

function V (t) = E[Nt]. We note that for an ordinary renewal process V (t) =∑∞k=1 F

?k(t),

and for a stationary renewal process V (t) = tµ . If there is a function v such that

V (t) =∫ t0v(s)ds, then v is called the renewal density. If the renewal process would

have been defined to also count the renewal at the origin, then the renewal function

would be V (t)+1. One can also define a renewal function for the inverse subordinator.

Given an inverse subordinator {τt}, we define its renewal function U by U(t) = E[τt].

The renewal function can be expressed as follows:

U(t) = E[τt] =∫ ∞

0

P (τt > x)dx =∫ ∞

0

P (Yx ≤ t)dx =∫ ∞

0

F ?x(t)dx

The expression on the right hand side might be hard to evaluate, but its Laplace-

Stieltjes transform is easily calculated:

U(s) =∫ ∞

0

e−st∫ ∞

0

F ?x(dt)dx =∫ ∞

0

F (s)xdx

=∫ ∞

0

e−xψ(s)dx =1

ψ(s)(4)

Thus there is a one-to-one correspondence between the renewal function and the

distribution of {τt}. This also correlates with the similar result for ordinary renewal


processes and their renewal functions. Define the factorial power n[k] for n, k ∈ N by:

n[k] =

n(n− 1) · · · (n− k + 1) for n ≥ k ≥ 1

1 for k = 0

0 for n < k, k ≥ 1.

Given a renewal process and its renewal function, moments of all orders can be calcu-

lated as stated in the following proposition, see [5].

Proposition 1. Let {Nt} be a renewal process with interarrival distribution F and let

V (t) =∑∞k=1 F

?k(t). If {Nt} is an ordinary renewal process then, for 0 ≤ s1 < t1 ≤

s2 < · · · < tn and k1, . . . , kn ∈ N \ {0} such that k1 + · · ·+ kn = k,

E[ n∏i=1

(Nti −Nsi)[ki]

]=

n∏i=1

ki! ·∫C

k∏j=1

V (dxj − xj−1), (5)

where C = {x0, . . . , xk;x0 = 0, si < xk0+···+ki−1+1 < · · · < xk0+···+ki≤ ti, i =

1, . . . , n, k0 = 0}. If {Nt} is stationary, then the proposition also holds with the first

factor of the rightmost product in equation (5) replaced by dx1µ .

A sketch of a proof: We can write Nt −Ns =∫(s,t]

N(dx) and

(Nt −Ns)k =∫

(s,t]k

k∏j=1

N(dxj).

Note that nk is the number of k-tuples of integers from 1 to n, and n[k] is the number

of k-tuples of integers such that no integers in the k-tuple are the same. Thus we can

write

(Nt −Ns)[k] =∫A

k∏j=1

N(dxj),

where A = {(x1 . . . xk) ∈ (s, t]k;xp 6= xq for p 6= q}. The renewal property is used in

the following:

E[ k∏j=1

N(dxj)]

= P (N(dx1) = 1, . . . , N(dxk) = 1)

= P (N(dx(1)) = 1)k∏j=2

P (N(dx(j)) = 1|N(dx(j−1)) = 1)

= P (N(dx(1)) = 1)k∏j=2

V (dx(j) − x(j−1)),


and the first factor equals V (dx(1)) and dx(1)

µ in the ordinary and stationary case,

respectively. Let Ai = {(yi1, . . . , yiki) ∈ (si, ti]ki ; yip 6= yiq for p 6= q} and Bi =

{(yi1, . . . , yiki); si < yi1 < · · · < yiki

≤ ti}. Thus, in the ordinary case,

E[ n∏i=1

(Nti −Nsi)[ki]]

= E[ n∏i=1

∫Ai

ki∏j=1

N(dyij)]

= E[ n∏i=1

ki!∫Bi

ki∏j=1

N(dyij)]

=n∏i=1

ki! · E[ ∫

C

k∏l=1

N(dxl)]

=n∏i=1

ki! ·∫C

k∏l=1

V (dxl − xl−1).

4. Inverse subordinators and Cox processes

An expression similar to (5) for the moments of {τt} can be obtained. First recall

the definition of a Cox process. Let {Nλt } be an inhomogeneous Poisson process on R+

with intensity measure λ. Let Λ be a random measure on R+. If the point process {Mt}

has the distribution of {Nλt } conditional on Λ = λ, then {Mt} is called a Cox process

directed by Λ. Note that if {Nt} is a Poisson process with constant intensity equal

to one and independent of Λ, then Mtd= N(Λ((0, t])), and {Mt} can be considered to

be a homogeneous Poisson process subjected to a random time change by the random

function Λ((0, t]). The interpretation of the Cox process as a time changed Poisson

process also describes how the points of the Cox process can be obtained from the

points of the Poisson process: If we let K(t) be the inverse of Λ((0, t]) and t1, t2, . . .

are the points of {Nt}, then K(t1),K(t2), . . . are the points of {Mt}.

Also define a slight generalization of the inverse subordinators: Let Y0 have the

distribution G on R+ and be independent of the subordinator {Yt} with Y1 ∼ F .

Define the process {Yt} by Yt = Yt + Y0. Let τt = inf(τ > 0 : Yτ > t), and call the

process {τt}t≥0 a general inverse subordinator. If Y0 ≡ 0 then we call {τt} an ordinary

inverse subordinator.

We will see in Proposition 4 that Y0 can be chosen so that the general inverse


subordinator {τt} has stationary increments, if µ = E[Y1] < ∞. The following

proposition is by Kingman [10] and Grandell [6].

Proposition 2. The Cox process {Mt} directed by Λ is a renewal process if and only

if Λ((s, t]) = τt − τs for all t > s, where {τt} is a general inverse subordinator.

We will only prove the easier if-part of the proposition. Only that part will be used in

theorem 1.

Proof. We note that Yt is the inverse of Λ((0, t]) = τt. If we use the representation

of {Mt} as a time changed Poisson process {Nt} with intensity one, then the points

of {Mt} are Y (t1), Y (t2), . . . , where t1, t2, . . . are the points of {Nt}. Since {Yt} is

a subordinator, Y (t1), Y (t2) − Y (t1), Y (t3) − Y (t2), . . . are independent and Y (t2) −

Y (t1), Y (t3)− Y (t2), . . . are furthermore equally distributed. Thus {Mt} is a renewal

process.

We can say more about the interarrival distribution of {Mt}. Let Z = Y (t2)− Y (t1) =

Y (t2)− Y (t1), and let ε ∼ Exp(1), independent of {Yt}. Z = Y (t2)− Y (t1)d= Y (t2 −

t1)d= Y (ε), so the interarrival distribution of {Mt} is thus compound exponential. The

Laplace-Stieltjes transform of the distribution of Z is given by

FZ(s) = E[e−sZ ] = E[E[e−sY (ε)|ε]] = E[e−ψ(s)ε] =1

1 + ψ(s), (6)

where ψ(s) is the Levy exponent of Y1. We now have the tools to prove the main

result:

Theorem 1. Let {τt} be an ordinary inverse subordinator with renewal function U(t).

Then, for 0 ≤ s1 < t1 ≤ s2 < · · · < tn and k1, . . . , kn ∈ N\{0} such that k1+ · · ·+kn =

k,

E[ n∏i=1

(τti − τsi)ki

]=

n∏i=1

ki! ·∫C

k∏j=1

U(dxj − xj−1) (7)

where C is as in Proposition 1. If {τt} is stationary, then the theorem also holds with

the change that the first factor of the rightmost product in equation (7) is replaced bydx1µ , but with the same U in the remaining factors as the ordinary inverse subordinator.


Proof. Define the random measure Λ on R+ by Λ((s, t]) = τt− τs for all t > s ∈ R+,

and let {Mt} be the Cox process directed by Λ. By Proposition 2, {Mt} is also a

renewal process. Write V (t) for its renewal function. Then

V (t) = E[Mt] = E[E[Mt|τt]] = E[τt] = U(t). (8)

Thus one can replace V (t) by U(t) in (5) when calculating the factorial moments of

{Mt}. As noted in [5], the factorial moments of the Cox process coincide with the

ordinary moments of its directing measure, and by the construction of the directing

measure the stated result follows.

A renewal theorem for the inverse subordinators can also be given following Bertoin

[2], Theorem I.21.

Proposition 3. If µ <∞, then U(t) ∼ tµ as t→∞.

Proof. Let {Mt} be a Cox process directed by {τt} as in Proposition 2, and V (t)

its renewal function. By (8), V (t) = U(t). An application of the renewal theorem for

renewal processes, see [5], provides the desired result.

Similar to renewal processes, the inverse subordinators can be delayed to become

stationary. This has been proved by different methods in [7] and [8]. We state the

result and provide a proof based on the connection with Cox processes.

Proposition 4. Let {τt} be a general inverse subordinator with Y0 ∼ G and Y1 =

Y1 − Y0 ∼ F and µ = E[Y1] <∞, where

ψ(s) = − log F (s) = δs+∫ ∞

0

(1− e−sx)ν(dx) and

G(x) =

1µ

(δ +

∫ x0

∫∞yν(dz)dy

)for x ≥ 0

0 for x < 0.(9)

Then {τt} has stationary increments.

Proof. By Theorem 1.4 in [6], a Cox process is stationary if and only if its directing

measure Λ has stationary increments. Therefore it suffices to check that the Cox

process {Mt} directed by {τt} is stationary. Its interarrival distribution is FZ given by

(6). The X1 of {Mt} can be decomposed into X1d= Y0+Z, with Y0 and Z independent,


since the inverse subordinator is delayed a time Y0 during which it is constant equal to

0. The Laplace-Stieltjes transform of the distribution H of X1 is H(s) = G(s)FZ(s),

where

G(s) =1µ

∫ ∞

0

e−sx(δ +

∫ ∞

x

ν(dy))dx =

1µ

(δ

s+∫ ∞

0

∫ y

0

e−sxdxν(dy))

=1µs

(δ +

∫ ∞

0

(1− e−sy)ν(dy))

=ψ(s)µs

. (10)

Combining (6) and (10), we get

H(s) = G(s)FZ(s) =ψ(s)µs

11 + ψ(s)

=1µs

(1− FZ(s)).

By (3), X1 thus has the right distribution to make {Mt} stationary.

LetWt be the excess of the renewal process and Cox process {Mt}, i.e. the time from

t to the next point of the process. When {Mt} is stationary, Wtd= X1

d= Y0 + Z. The

decomposition of the excess can be given the following interpretation: From any given

time t the inverse subordinator will remain constant a period which has the distribution

G. During this time no points in the Cox process will occur. After that time the inverse

subordinator starts anew and the distribution to the next point in the point process is

given by FZ . In the stationary case, we do not have to know G explicitly to calculate

E[Y0], if we use (2): E[X1] = E[Z2]2E[Z] . E[X1] = E[Y0] + EZ, and by straightforward

calculation, using e.g. (6), E[Z] = E[Y1] and E[Z2] = Var(Y1) + 2E[Y1]2. Collecting

and rearranging yields E[Y0] = Var(Y1)2EY1

.

The expression (7) may be hard to use in practice to calculate higher joint moments.

Nonetheless the results above show that the covariance of two increments of a stationary

inverse subordinator is a simple expression in the renewal function. Let {τt} be

stationary and let U(t) denote the renewal function of the corresponding ordinary

inverse subordinator. Also let 0 < r ≤ s < t.

Cov(τr, τt − τs) = E[τr(τt − τs)]− E[τr]E[τt − τs]

=∫ r

0

∫ t

s

U(dx− y)dy

µ− r

µ

t− s

µ

=1µ

∫ r

0

(U(t− y)− U(s− y))dy − r(t− s)µ2

.


Now consider the particular case where r = 1, s = n ≥ 1 and t = n + 1 and U has a

density u, such that U(t) =∫ t0u(s)ds. Also assume, for simplicity, that µ = 1. Then

the following approximation can be done:

Cov(τ1, τn+1 − τn) =∫ 1

0

(U(n+ 1− y)− U(n− y))dy − 1 ≈ u(n)− 1.

Given the distribution of the subordinator {Yt}, the distribution of its inverse is

given by P (τt ≤ x) = P (Yx > t). It may still be hard to find a closed form expression

of this distribution function. The tail probabilities for the ordinary inverse subordinator

can nonetheless be estimated. Only the case δ = 0 is interesting since if the drift δ is

positive then {Yx− δx} is non-negative and thus P (Yx ≤ t) = P (Yx− δx ≤ t− δx) = 0

for x > tδ . Let s ≥ 0. Then we have that

P (τt > x) = P (Yx ≤ t) = P (e−sYx ≥ e−st) ≤E[e−sYx

]e−st

= est−xψ(s).

By (1) the last expression has unique minimum as a function of s. If x is large enough

(x > tµ ), the s that minimizes the expression is non-zero and given by s = ψ′−1( tx ),

where ψ′−1 is the inverse of ψ′. Thus, for large enough x,

P (τt > x) ≤ exp(tψ′−1( tx )− xψ(ψ′−1( tx ))

). (11)

There is another result on the marginal distribution of {τt} that deserves mentioning.

This result can be found in [8] and [12], but we give a short proof based on identifying

Laplace transforms as probabilities.

Proposition 5. Let εs be exponentially distributed with mean 1s and independent of

{τt}. Then the Laplace-Stieltjes transform of the distribution of τ(εs) is given by:

E[e−uτ(εs)] = 1− uG(s)u+ ψ(s)

,

where G(s) is the Laplace-Stieltjes transform of the distribution of Y0.

Proof. Let εu be exponentially distributed with mean 1u and independent of εs

and {τt}. We note that for a non-negative random variable X independent of εu,


P (εu ≥ X) = E[P (εu ≥ X|X)] = E[e−uX ], the Laplace-Stieltjes transform of the

distribution of X. Thus we have, in the ordinary case,

E[e−uτ(εs)] = P (εu ≥ τ(εs)) = P (Y (εu) > εs) = 1− P (εs ≥ Y (εu))

= 1− E[e−sY (eεu)] = 1− E[E[e−sY (eεu)|εu]] = 1− E[e−ψ(s)eεu ]

= 1− u

u+ ψ(s).

Likewise, in the general case,

E[e−uτ(εs)] = 1− P (εs ≥ Y (εu))

= 1− P (εs ≥ Y (εu)|εs ≥ Y0)P (εs ≥ Y0)

= 1− P (εs ≥ Y (εu))P (εs ≥ Y0)

= 1− uG(s)u+ ψ(s)

,

where we have used the memorylessness of the exponential distribution.

5. Examples

The α-stable distribution on R+ has Levy exponent ψ(s) = sα with 0 < α < 1.

This gives a renewal density u(t) = 1/(Γ(α)t1−α) for the corresponding inverse stable

subordinator by inverting (4). Theorem 1 thus confirms the moment expressions in [4],

e.g. equation (18).

The main obstacle to use Theorem 1 is the possible difficulties in finding an expres-

sion for the renewal function. It is possible to find the renewal density not only for

the inverse stable subordinator, but also for the inverses of subordinators with inverse

gaussian and gamma distributed increments. In these two cases it is also possible to

delay the processes to obtain stationary versions, which is not possible in the stable

case.

For the inverse gaussian distribution, with probability density

f(x) =δ√

2πx3exp

(δγ − 1

2

(δ2

x+ γ2x

)), δ > 0, γ > 0,


and Levy exponent and Levy density, respectively,

ψ(s) = δ√γ2 + 2s− δγ

π(x) =δ√

2πx3exp

(−γ

2x

2

),

we get, by (9), a probability density of the delay Y0 by integrating π (µ = ψ′(0) = δγ )

g(t) =1µ

∫ ∞

t

π(x)dx = γ

√2πt

exp(−γ

2t

2

)− γ2 erfc

(γ

√t

2

)

Here erfc is the complementary error function defined by erfc(t) = 2√π

∫∞t

exp(−s2)ds.

We note that the density does not depend on the parameter δ. One obtains the renewal

density u(t) from its Laplace transform, which is equivalent to the Laplace-Stieltjes

transform of U(t), by rewriting (4):

U(s) =1

ψ(s)=

1

δ√γ2 + 2s− δγ

=γ

2δs+

1

δ√γ2 + 2s

+γ2

2δs√γ2 + 2s

⇒ {by [1] (29.3.1), (29.3.11) and (29.3.44)}

u(t) =γ

δ+

1δ√

2πtexp

(−γ

2t

2

)− γ

2δerfc

(γ

√t

2

)

The estimate (11) gives

P (τt > x) ≤ exp(−δ

2x2

2t+ δγx− γ2t

2

).

For the gamma distribution we have probability density, Levy exponent and Levy

density:

f(x) =αν

Γ(ν)xν−1e−αx, ν > 0, α > 0

ψ(s) = ν log(1 +

s

α

)π(x) =

ν

xe−αx

so, by (9), the density of the delay is

g(t) = αE1(αt),


where E1 the exponential integral defined by E1(t) =∫∞t

exp(−s)dss . As in the inverse

gaussian case the density only depends on one parameter. The renewal density is also

in the gamma case most easily obtained by first rewriting (4):

U(s) =1

ν log(1 + sα )

=α

νs

∫ 1

0

(1 +

s

α

)udu

=α

ν

∫ 1

0

(1s

1(1 + s

α )1−u+

1α

1(1 + s

α )1−u

)du

⇒ {by [1], (29.3.11), (29.2.6) and (6.5.2)}

u(t) =α

ν

∫ 1

0

du

Γ(u)(γ(u, αt) + (αt)u−1e−αt

),

where γ(u, t) =∫ t0su−1e−sds is the incomplete gamma function. We also have a tail

estimate by (11):

P (τt > x) ≤ exp(νx− αt− xν log

νx

αt

)=(αt

νx

)νxeνx−αt.

Acknowledgements

I am grateful to Jan Grandell for discussing Cox processes and renewal processes

with me, and also to Thomas Hoglund for careful reading of an earlier draft of this paper

and pointing out some errors. I also thank the anonymous referee, whose suggestions

helped to improve the article.

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