Uniform Convergence of Reversed Martingales€¦ · J. Theoret. Probab. Vol. 8, No. 2, 1995, (387-415) Preprint Ser. No. 21, 1992, Math. Inst. Aarhus Uniform Convergence of Reversed

J. Theoret. Probab. Vol. 8, No. 2, 1995, (387-415)

Preprint Ser. No. 21, 1992, Math. Inst. Aarhus

Uniform Convergence of Reversed

Martingales

GORAN PESKIR

A necessary and sufficient condition for the uniform convergence of a family of

reversed martingales converging to a degenerated limiting process is given. The

condition is expressed by means of regular convergence (in Hardy’s sense) of

corresponding means. It is shown that the given regular convergence is equivalent to

Hoffmann-Jørgensen’s eventually total boundedness in the mean which is necessary

and sufficient for the uniform law of large numbers. Analogous results are carried

out for families of reversed submartingales. By applying derived results several

convergence statements are obtained which extend those from the uniform law of

large numbers to the general reversed martingale case.

1. Introduction

Let f �n j n � 1 g be a sequence of independent identically distributed random functions

defined on a probability space (;F ; P ) with values in a measurable space (S;A) and a common

distribution law � , let T be a set, and let f : S � T ! R be an arbitrary function. Suppose

that the �-mean function associated to f :

M(t) =

ZSf(s; t) �(ds)

exists for all t 2 T , then in [3], [5], [14] and [15] one can find a series of necessary and sufficient

conditions for the following form of the law of large numbers:

(1.1)1

n

nXj=1

f(�j(!); t) M(t)

where the convergence is uniform over t 2 T , for all ! 2 outside some P -null set N from F .

Let us note if f(�j(!); � ) and M( � ) belongs to B(T ) for all j � 1 and ! 2 ; where B(T )denotes the Banach space of all bounded real valued functions on T relative to the sup-norm, then

(1.1) just states that the sequence of functions f f(�n; � ) j n � 1 g from into B(T ) satisfies

the strong law of large numbers in the Banach space B(T ) . A classical example which is covered

by this setting is the well-known Glivenko-Cantelli theorem. In its case the function f is defined

AMS 1980 subject classifications. Primary 60B12, 60F15, 60F17, 60F25, 60G42. Secondary 28A20.Key words and phrases: Reversed martingale, uniform convergence, Hardy’s convergence, totally bounded in the mean. [email protected]

1

on R�R with values in R by f(s; t) = 1H(s; t) , where H = f (s; t) 2 R�R j s � t g , and

it is very interesting and spectacular that the induced random functions f(�j ; � ) for j � 1 are

not necessarily measurable with respect to the Borel �-algebra on B(R) , as well as not separably

valued in B(R) , see [6]. Therefore the classical Banach space versions of the strong laws of large

numbers based on measurability of random functions in question and separability of a Banach range

space are too week to cover the first and the most natural example of the infinitely dimensional

law of large numbers. And in [6] one can find infinitely dimensional versions of the strong law

of large numbers which do not assume neither measurability nor separability, and which cover the

Glivenko-Cantelli theorem, as well as many other uniform laws of large numbers for stochastic

processes. It is well-known that the sequence:

Xn(t) =1

n

nXj=1

f(�j; t)

forms a reversed martingale relative to the associated permutation invariant �-algebras for every

t 2 T for which f( � ; t) 2 L1(�) . Moreover, the same is true for sequences:

Yn(t) =1

n

nXj=1

�f(�j; t)�M(t)

�2Zn(t) =

1

n� 1

nXj=1

hf(�j; t)� 1

n

nXk=1

f(�k; t)i2

for every t 2 T for which f( � ; t) 2 L2(�) . Therefore having all these facts in mind, it is very

natural to ask when does a family of reversed martingales (or submartingales) converge uniformly?

In [7] one can find such conditions for the uniform convergence of families of ordinary martingales

and submartingales, and it is indeed surprising that these conditions are very week compared to the

necessary and sufficient conditions for the uniform law of large numbers. The main reason for this

surprise lies in the well-known fact that reversed martingales and submartingales usually behave

much more regularly than ordinary ones. And in this paper we will establish such conditions for

the uniform convergence of reversed martingales and submartingales. We shall begin in the next

section by introducing the basic terminology and notation and presenting some preliminary results

needed in the rest of the paper. In particular, we shall summarize some elementary facts on regular

convergence (in Hardy’s sense) of double sequences. Then we shall obtain a series of necessary

and sufficient conditions for the a:s: and L1-convergence of supremum of a countable family of

reversed submartingales, provided that some mild limited condition on its means is satisfied, see

Theorem 3.1, Corollary 3.2 and Remark 3.3. It is remarkable, which is pointed out by J. Hoffmann-

Jørgensen in private communications, that these conditions actually describe regular convergence of

a double sequence of given means, introduced by G. H. Hardy in [4] and independently rediscovered

by F. Moricz in [9]. Moreover, one can easily deduce that these conditions confirm a conjecture of

J. Hoffmann-Jørgensen in [7] that considering uniform convergence, we have a case where ordinary

martingales behave more regularly than reversed ones. And using this result in the sequel we shall

derive a necessary and sufficient condition for the uniform convergence of a family of reversed

martingales (and submartingales) converging to a degenerated limiting process, which is indexed

2

by a separable topological space, see Theorem 4.1 and Theorem 4.3 together with Theorem 4.7

and Theorem 4.11, respectively. Furthermore, it will be shown that the given condition for the

uniform convergence of a family of reversed martingales is equivalent to the condition of eventually

total boundedness in the mean discovered for the uniform law of large numbers by J. Hoffmann-

Jørgensen in [5], see Theorem 4.11 and Remark 4.12. Applying derived results we shall obtain

several convergence statements which extend some of those in [5] to the general reversed martingale

case, see Corollary 4.2, Corollary 4.4 and Example 4.13. At the end in Example 4.14 we will see that

the convergence estimates given in Theorem 4.1 and Theorem 4.3 can not be improved in general.

2. Preliminary facts

Throughout this paper, let (;F ; P ) be a fixed probability space, and let f Fn j n � �1 gbe a given increasing sequence of sub-�-algebras of F with the intersection F�1 . If T is a

non-empty set, then RT denotes the set of all real valued functions defined on T , and B(T )denotes the set of all bounded functions from RT . For f 2 RT and A � T we put:

MA(f) = supt2A

f(t) and k f kA = supt2A

j f(t) j .

Then k f � g kT defines a metric on RT , not necessarily finite valued, but topologically

equivalent to the bounded metric arctan k f � g kT . It is well-known that B(T ) is a Banach

space, relative to the norm k � kT . A finite cover of T is a family = f D1; . . . ; Dn g of

non-empty subsets of T satisfying T =Snj=1Dj . The family of all finite covers of T will be

denoted by �(T ) . If (T; �) is a topological space, then C(T ) = C(T; �) denotes the set of all real

valued � -continuous functions defined on T , Usc (T ) = Usc (T; �) denotes the set of all real

valued upper � -semicontinuous functions defined on T , and Lsc (T ) = Lsc (T; �) denotes the

set of all real valued lower � -semicontinuous functions defined on T . If (T; �) is a pseudometric

space, then Cu(T ) = Cu(T; �) denotes the set of all real valued uniformly �-continuous functions

defined on T . It is easy to check that Cu(T; �) is a k � kT -closed subspace of RT .

Let (T; �) be a topological space, let t 2 T be a given point, and let Nt be the family of

all open neighborhoods of the point t . Let us recall that for given f 2 RT and G 2 Nt , the

lower, upper and absolute oscillation of f at t over G is respectively defined by:

W+G (t; f) = sup

s2G(f(t)� f(s))

W�G (t; f) = sup

s2G(f(s)� f(t))

WG(t; f) = max f W+G (t; f) ; W�

G (t; f) g = sups2G

j f(t)� f(s) j .

The lower, upper and absolute jump of f at t relative to � is respectively defined by:

@+(t; f) = @+� (t; f) = infG2Nt

W+G (t; f)

3

@�(t; f) = @�� (t; f) = infG2Nt

W�G (t; f)

@(t; f) = @� (t; f) = max f @+� (t; f) ; @�� (t; f) g = infG2Nt

WG(t; f) .

Clearly, we have:

(2.1) f is lower � -semicontinuous at t , if and only if @+� (t; f) = 0

(2.2) f is upper � -semicontinuous at t , if and only if @�� (t; f) = 0

(2.3) f is � -continuous at t , if and only if @� (t; f) = 0 .

Finally, for each f 2 RT we shall put:

�+(f) = �+� (f) = sup f @+� (t; f) j t 2 T g

��(f) = �� (f) = sup f @�� (t; f) j t 2 T g

�(f) = �� (f) = sup f @� (t; f) j t 2 T g .

Let us recall that a topological space T is called hereditarily separable, if every subspace of

T is separable. It is well-known that every separable pseudometric space is hereditarily separable,

but R2 with the product left (right) Sorgenfrey topology is an example of a separable normal

space which is not hereditarily separable. In our considerations on hereditarily separable topologies

the following statement will be useful:

(2.4) If �1 and �2 are hereditarily separable topologies on a set T such that �1 has

a countable base, then there is a hereditarily separable topology � on T finer than

�1 and �2 .

Indeed, if we take a countable base B1 of the topology �1 which is closed under the formations

of finite intersections, then it is easy to verify that the family:

B = f B1 \G2 j B1 2 B1 ; G2 2 �2 g

is a base of a desired topology � .

Let f Xn j �1 < n � �1 g be a sequence of integrable random variables defined on

(;F ; P ) . Let us recall that the family f Xn ; Fn j �1 < n � �1 g is called a reversed

martingale, submartingale or supermartingale, if Xn is Fn-measurable and Xn = Ef Xn+1 jFn g , Xn � Ef Xn+1 j Fn g or Xn � Ef Xn+1 j Fn g , for all �1 < n � �2 ;respectively. We shall refer the reader to [1] for basic properties and convergence statements of

reversed martingales and submartingales which we are going to use mainly implicitly in the rest

of this paper. Let us recall that an arbitrary function Z : ! R is called a real valued random

element on (;F ; P ) : Let f Zn j n � 1 g be a sequence of real valued random elements defined

4

on (;F ; P ) . In order to describe certain non-measurable random phenomenons which occur

naturally in our next considerations, we shall introduce the following convergence notions:

(i) Zn ! 0 (a:s:) , if there exists a P -null set N 2 F such that Zn(!)! 0 ; 8! 2 nN(ii) Zn ! 0 (a:s:)� , if there exist random variables Dn defined on (;F ; P ) , satisfying

Dn ! 0 (a:s:) and jZnj � Dn for all n � 1

(iii) Zn ! 0 (P �) , if P �f jZnj � " g ! 0 ; 8" > 0

(iv) Zn ! 0 (P�) , if P�f jZnj � " g ! 0 ; 8" > 0

(v) Zn ! 0 (L1)� , if E�jZnj ! 0

(vi) Zn ! 0 (L1)� , if E�jZnj ! 0

where P � and P� denotes the outer and inner P -measure, and E� and E� denotes the upper

and lower P -integral. Note if every function Zn is measurable for n � 1 , then the convergence

notion in (i) and (ii) coincides with the notion of P -almost surely convergence ( in next denoted

by (a:s:) ), the convergence notion in (iii) and (iv) coincides with the notion of convergence in

P -probability ( in next denoted by (P ) ), and the convergence notion in (v) and (vi) coincides with

the notion of convergence in L1(P ) ( in next denoted by (L1) ) of the given sequence of random

variables f Zn j n � 1 g , respectively. For more informations on the convergence notions in

(i)-(vi) we shall refer the reader to [11] and [12]. It is easy to establish that we have:

(a:s:)� ) (a:s:)

+ +(2.5) (P �) ) (P�)

* *(L1)� ) (L1)�

and no other implication holds in general. Let f Zn j n � 1g and fVn j n � 1 g be two

sequences of random elements, and let (c) denote either of the following convergence notions:

(a:s:); (a:s:)�; (P �); (L1)� , but neither (P�) nor (L1)� , then it is easy to verify that we have:

(2.6) Zn ! 0 (c) and Vn ! 0 (c) ) Zn + Vn ! 0 (c)

(2.7) Zn ! 0 (P�) and Vn ! 0 (P �) ) Zn + Vn ! 0 (P�)

(2.8) Zn ! 0 (L1)� and Vn ! 0 (L1)� ) Zn + Vn ! 0 (L1)� .

Let us recall that a double sequence of real numbers A = f ank j n; k � 1 g is convergent

(in Pringsheim’s sense) to the limit A , if 8" > 0 ; 9p" � 1 such that 8n; k � p" we have

j A� ank j < " . In this case we shall write A = limn;k!1 ank . And following G. H. Hardy [4]

we say that a double sequence of real numbers A = f ank j n; k � 1 g is regularly convergent

5

(in Hardy’s sense) to the limit A , if all limits:

limn!1 ank ; lim

k!1ank ; lim

n;k!1ank

exist, for all n; k � 1 , and the last one is equal to A . In this case we necessarily have:

limn!1 lim

k!1ank = lim

k!1limn!1 ank = lim

n;k!1ank = A .

In the results below we shall need the following lemma:

Lemma 2.1

Let E = f Enk j n; k � 1 g be a double sequence of real numbers satisfying the following

two conditions:

(i) En1 � En2 � En3 � . . . ; for all n � 1

(ii) E1k � E2k � E3k � . . . ; for all k � 1 .

Let En1 = limk!1

Enk and E1k = limn!1Enk for n; k � 1 . Then the following five statements

are equivalent:

(1) E is regularly convergent (in Hardy’s sense)

(2) E is convergent (in Pringsheim’s sense)

(3) �1 < limk!1

E1k = limn!1En1 < +1

(4) 8" > 0 ; 9p" � 1 such that 8n;m; k; l � p" we have j Enk � Eml j < "

(5) 8" > 0 ; 9p" � 1 such that Ep"1 � E1p" < " .

Proof. The proof is straight forward and we shall leave the details to the reader.

3. Total boundedness in the mean

The essential conclusions in the proofs of the main results in [7] are provided by using Lemma

V-2-9 in Neveu’s book [10], which gives a mild sufficient condition for the a:s:-convergence of

supremum of a countable family of ordinary submartingales (f X in ; Fn j n � 1 g ; i 2 N) . A

close look into its proof shows that it has mainly due to the fact that the function:

(3.1) (n; k) 7�! E( sup1�i�k

X in)

is non-decreasing in each variable. This is no longer true in the general reversed submartingale case,

where the function from (3.1) is still non-decreasing in k , but it is decreasing in n , and it is easy

6

to see that Neveu’s lemma can fail in this case in general. It turns out that this irregularity is the

main technical detail which makes the uniform convergence of families of reversed submartingales

much more harder to establish than for families of ordinary ones. Our next theorem gives an

analogue of Neveu’s lemma for reversed submartingales.

Theorem 3.1

Let (f X in ; Fn j �1 < n � �1 g j i 2 N) be a countable family of reversed submartingales

and let X i�1 denote the a:s: limit of X in , as n ! �1 , for all i 2 N . If the following

condition is satisfied:

(i) 8" > 0 ; 9p" � 1 such that 8n;m; k; l � p" we have:

j E( sup1�i�k

X i�n)� E( sup

1�j�lXj�m) j < "

then we have:

(ii) 9k0 � 1 such that �1 < infn��1

E( sup1�i�k0

X in) � inf

n��1E(sup

i2NX in) < +1

(iii) supi2N

X in �! sup

i2NX i�1 (a:s:) and (L1) , as n ! �1 .

Conversely, if we have convergence in P -probability in (iii) together with the following condition:

(iv) �1 < infn��1

E(supi2N

X in) < +1

then (i) holds.

Proof. First suppose that (i) holds, then letting k ! 1 in (i) and using the monotone

convergence theorem we may deduce:

j E(supi2N

X i�p")� E( sup

1�j�p"Xj�p") j � "

Hence we see that the last inequality in (ii) must be satisfied, and moreover one can easily verify

that for every M subset of N the family:

(1) f supi2M

X in ; Fn j �1 < n � p" g

forms a reversed submartingale. Therefore by the reversed submartingale convergence theorem we

may conclude:

(2) supi2M

X in �! XM

�1 (a:s:)

(3) E(supi2M

X in) �! E(XM

�1)

7

(4) supi2M

X in �! XM

�1 (L1) ; if infn��1E(sup

i2MX in) > �1

as n ! �1 . Note if M is a finite subset of N , then XM�1 = sup

i2MX i�1 a:s: . Therefore

letting n ! 1 in (i) and using (3) we may obtain:

j E( sup1�i�p"

X i�1)� E( sup

1�j�p"Xj�p") j � " .

Hence sup1�i�p"

X i�1 2 L1(P ) , and thus by (3) we may conclude:

infn��1E( sup

1�i�p"X in) = E( sup

1�i�p"X i�1) > �1 .

This completes the proof of (ii). Moreover infn��1E(sup

i2NX in) > �1 , and thus by (4) we have:

(5) supi2N

X in �! XN�1 (L1)

as n! �1 . And in order to establish (iii) let us now note that by (2) and (5) it is enough to

show that XN�1 = supi2N

X i�1 a:s: . Since the inequality XN�1 � sup

i2NX i�1 a:s: follows straight

forward and XN�1 2 L1(P ) , for this last conclusion it is sufficient to show that:

(6) EXN�1 = E(supi2N

X i�1) .

And in order to deduce (6) let us note that the double sequence E = f Enk j n; k � 1 g defined by:

(7) Enk = E( sup1�i�k

X i�n)

satisfies conditions (i) and (ii) in Lemma 2.1, and moreover condition (i) in Theorem 3.1 is exactly

condition (4) in Lemma 2.1. Therefore by (3) in Lemma 2.1 and the monotone convergence

theorem we may conclude:

EXN�1 = limn!1 lim

k!1E( sup

1�i�kX i�n) = lim

k!1limn!1E( sup

1�i�kX i�n) = E(sup

i2NX i�1) .

Thus (6) is established, and the proof of (iii) is complete.

Now suppose that (iv) holds and that we have convergence in P -probability in (iii). Then by (1)

and the second inequality in (iv) the family f supi2N

X in ; Fn j �1 < n � n1 g is a reversed sub-

martingale, for some n1 � �1 , which by the first inequality in (iv) satisfies:

infn�n1

E(supi2N

X in) > �1 .

8

Therefore by (2) and (4) statement (iii) follows straight forward. Moreover, by (iii) and the

monotone convergence theorem we may conclude:

�1 < limn!1 lim

k!1E( sup

1�i�kX i�n) = lim

k!1limn!1E( sup

1�i�kX i�n) <1 .

Hence (i) follows directly by applying the implication (3) ) (4) in Lemma 2.1 on the double

sequence E = f Enk j n; k � 1 g defined by (7) above, and the proof is complete.

Corollary 3.2

Let (f X in ; Fn j �1 < n � �1 g j i 2 N) be a countable family of reversed submartingales

and let X i�1 denote the a:s: limit of X in , as n! �1 , for all i 2 N . Suppose that:

�1 < infn��1E(sup

i2NX i

n) < +1

and let Enk = E( sup1�i�k

X i�n) for all n; k � 1 . Then the family:

f supi2N

X in ; Fn j �1 < n � n1 g

is a reversed submartingale, for some n1 � �1 , and supi2N

X in converges P -almost surely and in

L1(P ) , as n ! �1 . Moreover, we have:

supi2N

X in �! sup

i2NX i�1 (a:s:) and (L1)

as n ! �1 , if and only if the double sequence E = f Enk j n; k � 1 g is regularly convergent

(in Hardy’s sense).

Remark 3.3 Let us note that under the hypotheses in Corollary 3.2, the double sequence

E = f Enk j n; k � 1 g satisfies conditions (i) and (ii) in Lemma 2.1, so it is regular convergent

if and only if either of statements (2)-(5) in Lemma 2.1 is satisfied. Notice that we have:

En1 = E(supi2N

X i�n) and E1k = E( sup1�i�k

X i�1).

In order to extend the result in Theorem 3.1 to not necessarily countable families of reversed

submartingales, we shall introduce the following definition: Let T be a non-empty set, let

f Xn(t) ; Fn j �1 < n � �1 g

be a reversed submartingale for t 2 T , and let X�1(t) denote the a:s:-limit of Xn(t) , as

n ! �1 , for all t 2 T . Let D = f di j i � 1 g be a countable subset of T , and let

Dn = f d1; . . . ; dn g , for all n � 1 . Then the family of reversed submartingales:

9

( f Xn(t) ; Fn j �1 < n � �1 g j t 2 T )

is said to be totally bounded in P-mean relative to D , if either of the following nine equivalent

conditions is satisfied, see Remark 3.3 and Lemma 2.1:

(i) The double sequence f EMDk(X�n) j n; k � 1 g is regularly convergent (in Hardy’s

sense)

(ii) The double sequence f EMDk(X�n) j n; k � 1 g is convergent (in Pringsheim’s sense)

(iii) �1 < limk!1

EMDk(X�1) = lim

n!1EMD(X�n) < +1 ;

(iv) 8" > 0 ; 9p" � 1 such that 8n;m; k; l � p" we have:

j EMDk(X�n)� EMDl

(X�m) j < "

(v) 8" > 0 ; 9p" � 1 such that 8m � n � p" and 8k � l � p" we have:

EMDk(X�n)� EMDl

(X�m) < "

(vi) 8" > 0 ; 9p" � 1 such that 8m � n � p" and 8l � k � p" we have:

j EMDk(X�n)� EMDl

(X�m) j < "

(vii) 8" > 0 ; 9p" � 1 such that 8n; k � p" we have:

j EMDk(X�n)� EMDp"

(X�p") j < "

(viii) 8" > 0 ; 9p" � 1 such that 8n; k � p" we have:

EMDk(X�p")� EMDp"

(X�n) < "

(ix) 8" > 0 ; 9p" � 1 such that:

EMD(X�p")� EMDp"(X�1) < " .

In this case the limit of the double sequence f EMDk(X�n) j n; k � 1 g is equal to EMD(X�1) ,

and we have:

limk!1

limn!1EMDk

(X�n) = EMD(X�1) = limn!1 lim

k!1EMDk

(X�n) .

Moreover then by Theorem 3.1 we have:

MD(Xn) �!MD(X�1) (a:s:) and (L1)

as n ! �1 .

4. The reversed martingale convergence theorem

In this section we shall extend the result in Theorem 3.1 to families of reversed martingales and

submartingales indexed by a separable topological space. We first consider the reversed submartin-

10

gale case in Theorem 4.1, and then we shall pass to its martingale version in Theorem 4.3. The

results in Theorem 4.1 and Theorem 4.3 will be completed by forthcoming results in Theorem 4.7

and Theorem 4.11, respectively.

Theorem 4.1

Let ( f Xn(t) ; Fn j �1 < n � �1 g j t 2 T ) be a family of reversed submartingales

indexed by a separable topological space T , let D be a countable dense subset of T , and let

L be a map from T into R . Suppose that:

(i) Xn(t) ! L(t) (a:s:) , as n ! �1 , for all t 2 T

(ii) The family of reversed submartingales

(f (Xn(t)� L(t))+ ; Fn j �1 < n � �1 g j t 2 T )

is totally bounded in P -mean relative to D

and either of the following two conditions is satisfied:

(iii) L 2 Usc (T )

(iv) T is hereditarily separable and (ii) holds for each countable D subset of T .

Then there exists a sequence of random variables f Vn j n � �1 g satisfying:

(v) Vn ! 0 (a:s:) and (L1) , as n ! �1(vi) k (Xn(!)� L)+ kT � �+

n (!) + Vn(!) ; 8! 2

where �+n (!) = �+(Xn(!)) = supf @+(t; Xn(!)) j t 2 T g . In particular, if �+

n ! 0 (c) , then

k (Xn � L)+ kT ! 0 (c) , as n! �1 , where (c) denotes either of the following convergence

notions: (a:s:); (a:s:)�; (P �); (P�); (L1)�; (L1)� . Conversely, if (i) is satisfied and:

k (Xn � L)+ kT ! 0 (P�)

as n ! �1 , then (ii) holds for an arbitrary countable D subset of T satisfying

infn��1E k (Xn � L)+ kD <1 .

Proof. Let f; g 2 RT be arbitrary functions, let t be a point in T , and let Nt be the

family of all open neighborhoods of t . Since D is dense in T , then for each G; H 2 Nt there

exists d 2 G \ H \ D , and consequently we may deduce:

f(t)� g(t) = (f(t)� f(d)) + (g(d)� g(t)) + (f(d)� g(d)) �

� W+G (t; f) +W�

H (t; g)+ k (f � g)+ kD .

Taking infimums over all G;H 2 Nt we obtain:

11

(f(t)� g(t))+ � @+(t; f) + @�(t; g)+ k (f � g)+ kDAnd taking supremum over all t 2 T we may conclude:

k (f � g)+ kT � �+(f) + ��(g)+ k (f � g)+ kDfor all f; g 2 RT . In particular, by (2.2) we have:

k (f � g)+ kT � �+(f)+ k (f � g)+ kDfor all f 2 RT and all g 2 Usc (T ) . Therefore (iii) implies:

k (Xn(!)� L)+ kT � �+(Xn(!))+ k (Xn(!)� L)+ kDfor all ! 2 and all n � �1 . Let us note that the family:

( f (Xn(t)� L(t))+ ; Fn j �1 < n � �1 g j t 2 D )

is a countable family of reversed submartingales which by (i) satisfies:

(Xn(t)� L(t))+ �! 0 (a:s:)

as n ! �1 , for all t 2 D . But then by (ii) and Theorem 3.1 we may conclude:

Vn := k (Xn(t)� L(t))+ kD �! 0 (a:s:) and (L1)

as n ! �1 . Consequently, we see that (i),(ii) and (iii) imply (v) and (vi).

Next suppose that (i), (ii) and (iv) hold. Let � be the given hereditarily separable topology

on T . Note that:

�(t0; t00) = arctan j L(t0)� L(t00) j

defines a separable pseudometric on T such that L 2 Cu(T; �) . Let �� denote the topology

generated by � , then by (2.4) there exists a hereditarily separable topology � on T which is

finer than � and �� . Let D be a countable �-dense subset of T . Now let us return to the

beginning of the proof with the new separable topology � on T and let us note that:

L 2 Cu(T; �) � C(T; �) � Usc (T; �) .

Thus (iii) holds if we replace � by �� . Since � � � , we have:

@+� (t; f) � @+� (t; f)

for all f 2 RT and all t 2 T . Therefore we may conclude:

�+� (Xn(!)) � �+

� (Xn(!))

for all ! 2 and n � 1 . Consequently, (v) and (vi) follow straight forward by the first part of

12

the proof. Finally note that the convergence statement, stated after statement (vi) in the theorem,

follows directly by (2.6)-(2.8).

Conversely, if k (Xn � L)+ kT ! 0 (P�) , as n! �1 , then obviously:

k (Xn � L)+ kD ! 0 (P )

as n ! �1 , for every countable D subset of T . But then (ii) follows by Theorem 3.1,

and the proof is complete.

The next corollary is an easy consequence of the equivalence statement obtained in the previous

theorem, and we shall leave its verification to the reader.

Corollary 4.2

Let ( f Xn(t) ; Fn j �1 < n � �1 g j t 2 T ) be a family of reversed submartingales

indexed by a separable topological space T , let D be a countable dense subset of T , and let

L be a map from T into R . Suppose that:


(ii) k (Xn(!) � L)+ kT ! 0 (P�) , as n ! �1


(iii) infn��1E k (Xn � L)+ kD <1 and L 2 Usc (T )

(iv) infn��1

E k (Xn � L)+ kD <1 for each countable D subset of T , and T is

hereditarily separable.

If �+(Xn) ! 0 (c) , then k (Xn � L)+ kT ! 0 (c) , as n ! �1 , where (c) denotes either

of the following convergence notions: (a:s:); (a:s:)�; (P �); (P�); (L1)�; (L1)� . In particular, if Tis hereditarily separable, Xn(!; � ) 2 Lsc(T ) for a:a: ! 2 and all n � �1 , and:

infn��1E

� k (Xn � L)+ kT <1

then k (Xn � L)+ kT ! 0 (P�) , implies k (Xn � L)+ kT ! 0 (a:s:)� and (L1)� , as n! �1:

Theorem 4.3

Let (f Xn(t) ; Fn j �1 < n � �1 g j t 2 T ) be a family of reversed martingales indexed

by a separable topological space T , let D be a countable dense subset of T , and let L be a

map from T into R . Suppose that:


13

(ii) The family of reversed submartingales

(f j Xn(t)� L(t) j ; Fn j �1 < n � �1 g j t 2 T )

is totally bounded in P -mean relative to D


(iii) L 2 C(T )(iv) T is hereditarily separable and (ii) holds for each countable D subset of T .

Then there exists a sequence of random variables f Vn j n � �1 g satisfying:

(v) Vn ! 0 (a:s:) and (L1) , as n ! �1(vi) k Xn(!)� L kT � �n(!) + Vn(!) ; 8! 2

where �n(!) = �(Xn(!)) = supf @(t; Xn(!)) j t 2 T g . In particular, if �n ! 0 (c) , then

k Xn � L kT ! 0 (c) , as n ! �1 , where (c) denotes either of the following convergence

notions: (a:s:); (a:s:)�; (P �); (P�); (L1)�; (L1)� . Conversely, if (i) is satisfied and:

k Xn � L kT ! 0 (P�)

as n ! �1 , then (ii) holds for an arbitrary countable D subset of T satisfying

infn��1

E k Xn � L kD <1 .

Proof. In the proof of Theorem 4.1 we have established the following inequality:

(f(t)� g(t))+ � @+(t; f) + @�(t; g)+ k (f � g)+ kDfor all f; g 2 RT and all t 2 T . Hence one can easily deduce that we have:

k f � g kT � �(f) + k f � g kDfor all f 2 RT and all g 2 C(T ) . Therefore (iii) implies:

k Xn(!)� L kT � �(Xn(!))+ k Xn � L kDfor all ! 2 and all n � �1 . Let us note that the family:

( f j Xn(t)� L(t) j ; Fn j �1 < n � �1 g j t 2 D )

is a countable family of reversed submartingales which by (i) satisfies:

j Xn(t)� L(t) j ! 0 (a:s:)

as n ! �1 , for all t 2 D . Hence by (ii) and Theorem 3.1 we may conclude:

14

Vn := k Xn � L kD ! 0 (a:s:) and (L1)

as n ! �1 . Consequently, we see that (i), (ii) and (iii) imply (v) and (vi).

The rest of the proof is very similar to the corresponding last part in the proof of Theorem 4.1,

and we shall leave the details to the reader.

Similarly to Corollary 4.2, the next corollary is an easy consequence of the equivalence

statement obtained in the previous theorem, and we shall leave its verification to the reader.

Corollary 4.4


by a separable topological space T , let D be a countable dense subset of T , and let L be a

map from T into R . Suppose that:


(ii) k Xn � L kT ! 0 (P�) , as n ! �1


(iii) infn��1E k Xn � L kD <1 and L 2 C(T )

(iv) infn��1E k Xn � L kD <1 for each countable D subset of T , and T is

hereditarily separable.

If �(Xn) ! 0 (c) , then k Xn � L kT ! 0 (c) , as n ! �1 , where (c) denotes either of

the following convergence notions: (a:s:); (a:s:)�; (P �); (P�); (L1)�; (L1)� . In particular, if T is

hereditarily separable, Xn(!; � ) 2 C(T ) for a:a: ! 2 and all n � �1 , and:

infn��1

E� k Xn � L kT <1

then k Xn � L kT ! 0 (P�) , implies k Xn � L kT ! 0 (a:s:)� and (L1)� , as n! �1:

We shall continue our considerations by exploring condition (ii) in Theorem 4.1 and Theorem

4.3, in order to obtain an equivalent form which is more suitable for applications. The main results

in this direction are established in Theorem 4.7 and Theorem 4.11 below, and they offer a convenient

criterion for that condition. We shall use X�n to denote (Xn)

� for n � 1 , respectively.

Proposition 4.5

Let (f Xn(t) ; Fn j �1 < n � �1 g j t 2 T ) be a family of reversed submartingales

indexed by a set T , let D be a countable subset of T , and suppose that the following two

conditions are satisfied:

15

(i) Xn(t) ! 0 (a:s:) , as n ! �1 , 8t 2 T

(ii) 8" > 0 ; 9 = f D1; . . . ; Dm" g 2 �(D) ; 9d1 2 D1; . . . ; dm" 2 Dm" such that

8N � �1 ; 9n" � N and 91; . . . ;m" 2 L1(P ) satisfying:

(1) (Xn"(t)�Xn"(dj))+ � j ; 8t 2 Dj ; 8j = 1; . . . ;m"

(2) Pf max1�j�m"

Ef j j F�1 g > " g < " .

Then the family of reversed submartingales:

(f X+n (t) ; Fn j �1 < n � �1 g j t 2 T )

is totally bounded in P -mean relative to D .

Proof. Let us first note that the family f EfX+n (t) j F�1 g ; Fn j �1 < n � �1 g forms

a reversed submartingale, for every t 2 T . Since obviously infn��1E[X+n (t) ] > �1 , then

f X+n (t) j n � �1 g is uniformly integrable for every t 2 T , and therefore we have:

EfX+n (t) j F�1 g ! 0 (a:s:) and (L1)

as n! �1 , for all t 2 T . Thus for given " > 0 and t 2 T , there exists n";t � �1 such that:

(1) j EfX+n (t) j F�1 g j < "

for all n � n";t . By (ii) we can find = f D1; . . . ; Dm" g 2 �(D) and d1 2 D1; . . . ; dm" 2 Dm"

such that 8N � �1 ; 9n" � N and 91; . . . ;m" 2 L1(P ) satisfying (1) and (2) in (ii). In

particular for N = min f n";d1; . . . ; n";dm" g there exists n" � N and 1; . . . ;m" 2 L1(P )satisfying (1) and (2) in (ii). Hence we may conclude:

X+n (t) � Ef X+

n"(t) j Fn g � Ef (Xn"(t)�Xn"(dj))+ j Fn g +

+ Ef X+n"(dj) j Fn g � Ef j j Fn g+ Ef X+

n"(dj) j Fn g

for all n � n" and all t 2 Dj with j = 1; . . .m" . Taking supremum over all t 2 D and

using (1) we get:

lim supn!�1

k X+n kD � max

1�j�m"

Ef j j F�1 g+ max1�j�m"

Ef X+n"(dj) j F�1 g �

� max1�j�m"

Ef j j F�1 g+ " .

Hence by (2) we easily obtain Pf lim supn!�1 k X+n kD > 2" g < " , for all " > 0 . Thus

we may conclude lim supn!�1 k X+n kD = 0 a:s: . By (1) in (ii) one can easily deduce

E k X+n" kD <1 , and therefore the family f k X+

n kD ; Fn j �1 < n � n" g forms a non-

negative reversed submartingale which converges P -almost surely to zero, as n! �1 . But then

E k X+n kD ! 0 , as n ! �1 , and by (5) in Lemma 2.1 and Remark 3.3 we may conclude

16

that the family of reversed submartingales (f X+n (t) ; Fn j �1 < n � �1 g j t 2 T ) is totally

bounded in P -mean relative to D . This fact completes the proof.

Proposition 4.6

Let (f Xn(t) ; Fn j �1 < n � �1 g j t 2 T ) be a family of reversed submartingales indexed

by a set T , let D be a countable subset of T , and suppose that the following condition is satisfied:

(i) Xn(t) ! 0 (a:s:) , as n ! �1 , 8t 2 T .

If the family of reversed submartingales:

( f X+n (t) ; Fn j �1 < n � �1 g j t 2 T )

is totally bounded in P -mean relative to D , then we have:

(ii) 8" > 0 ; 8 = fD1; . . . ; Dmg 2 �(D) ; 8d1 2 D1; . . . ; dm 2 Dm and 8N � �1 ,

9n" � N and 91; . . . ;m 2 L1(P ) satisfying:

(1) (Xn"(t)�Xn"(dj))+ � j ; 8t 2 Dj ; 8j = 1; . . . ;m

(2) E( max1�j�m

j) < " .

Proof. Let us first note that by (5) in Lemma 2.1 and Remark 3.3 the family of reversed

submartingales ( f X+n (t) ; Fn j �1 < n � �1 g j t 2 T ) is totally bounded in P -mean relative

to D , if and only if E k X+n kD ! 0 , as n! �1 . Let " > 0 ; = f D1; . . . ; Dm g 2 �(D)

and d1 2 D1; . . . ; dm 2 Dm be given, then we have:

(Xn(t)�Xn(dj))+ � X+

n (t) +X�n (dj)

for all n � �1 , all t 2 Dj and all j = 1; . . . ; m . Hence we find:

supt2Dj

(Xn(t)�Xn(dj))+ � k X+

n kDj+ X�

n (dj)

for all n � �1 and all j = 1; . . . ; m . Since we have infn��1E[ X�n (t) ] > �1 , then

f X�n (t) j n � �1 g is uniformly integrable for every t 2 T , and therefore we may deduce that

f max1�j�m

X�n (dj) j n � �1 g is uniformly integrable. Using this fact we may conclude that:

E( max1�j�mX

�n (dj) )! 0

as n ! �1 . Thus for given N � �1 , there exists n" � N satisfying:

E k X+n" kD + E( max

1�j�mX�n"(dj) ) < "

17

and the proof follows easily by putting:

j = k X+n"kDj

+ X�n"(dj)

for all j = 1; . . . ;m .

Theorem 4.7

Let (f Xn(t) ; Fn j �1 < n � �1 g j t 2 T ) be a family of reversed submartingales indexed

by a set T , let D be a countable subset of T , and suppose that the following condition is satisfied:

(i) Xn(t) ! 0 (a:s:) , as n ! �1 ; 8t 2 T .

Then the following three statements are equivalent:

(1) 8" > 0 ; 9 = f D1; . . . ; Dm" g 2 �(D) ; 9d1 2 D1; . . . ; dm" 2 Dm" such that


(i) (Xn"(t)�Xn"(dj))+ � j ; 8t 2 Dj ; 8j = 1; . . . ; m"

(ii) Pf max1�j�m"

Ef j j F�1 g > " g < "

(2) 8" > 0 ; 9 = f D1; . . . ; Dm" g 2 �(D) ; 9d1 2 D1; . . . ; dm" 2 Dm" such that


(i) (Xn"(t)�Xn"(dj))+ � j ; 8t 2 Dj ; 8j = 1; . . . ; m"

(ii) E( max1�j�m"

j) < "

(3) The family of reversed submartingales:

(f X+n (t) ; Fn j �1 < n � �1 g j t 2 T )

is totally bounded in P-mean relative to D .

Proof. The implications (1) ) (3) and (3) ) (2) follow by Proposition 4.5 and

Proposition 4.6, respectively. In order to prove the implication (2) ) (1) take " > 0 , then

9 = f D1; . . . ; Dm" g 2 �(D) and 9d1 2 D1; . . . ; dm" 2 Dm" , such that 8N � �1 ; 9n" � Nand 91; . . . ;m" 2 L1(P ) satisfying (i) in (2) and:

E( max1�j�m"

j) < "2 .

Hence by Markov’s inequality we may conclude:

Pf max1�j�m"

Ef j j F�1 g > " g � Pf Ef max1�j�m"

j j F�1 g > " g � 1

"E( max

1�j�m"

j) < " .

18

Thus (i) and (ii) in (1) are fulfilled, and the proof is complete.

Remark 4.8 Under the hypotheses in Theorem 4.7, suppose that the �-algebra F�1 is

degenerated, i.e. P (F ) 2 f0; 1g ; 8F 2 F�1 . Then Ef j j F�1 g = E(j) , for all

j = 1; . . . ; m" , and condition (ii) in (1) is equivalent to the following condition:

(1) max1�j�m"

E(j) � "

provided that " � 1 . Also note if ( f Xn(t) ; Fn j �1 < n � �1 g j t 2 T ) is a family

of reversed submartingales indexed by a set T , and L is a map from T into R satisfying

Xn(t) ! L(t) (a:s:) , as n ! �1 , 8t 2 T , and if we put Yn(t) = Xn(t) � L(t) , for all

n � �1 and all t 2 T ; then ( f Yn(t) ; Fn j �1 < n � �1 g j t 2 T ) becomes a family

of reversed submartingales satisfying hypotheses in Theorem 4.7. Hence we see that Theorem 4.7

offers criterions for condition (ii) in Theorem 4.1.

Now we shall pass to the martingale versions of the preceding three results. We could see

in Theorem 4.11 and Remark 4.12 below, that our condition of total boundedness in the mean is

actually equivalent to Hoffmann-Jørgensen’s condition of eventually total boundedness in the mean,

which is necessary and sufficient for the uniform law of large numbers, see [5].

Proposition 4.9


by a set T , let D be a countable subset of T , let L be a map from T into R ; and suppose

that the following two conditions are satisfied:

(i) Xn(t)! L(t) (a:s:) , as n ! �1 ; 8t 2 T

(ii) 8" > 0 ; 9n" � �1 ; 9 = f D1; . . . ; Dm" g 2 �(D) and 91; . . . ;m" 2 L1(P )satisfying:

(1) j Xn"(t0)�Xn"(t

00) j � j ; 8t0; t00 2 Dj ; 8j = 1; . . . ; m"

(2) Pf max1�j�m"

Ef j j F�1 g > " g < " .

Then the family of reversed submartingales:



Proof. For given " > 0 , there exists n" � �1 ; = f D1; . . . ; Dm" g 2 �(D) and

1; . . . ;m" 2 L1(P ) satisfying (1) and (2) in (i). For each j = 1; . . . ; m" choose a point dj in

19

Dj , then for given t 2 Dj we have:

j Xn"(t) j � j Xn"(t)�Xn"(dj) j + j Xn"(dj) j �

� j + j Xn"(dj) j �m"Xj=1

�j+ j Xn"(dj) j

�.

Since the map on the right side above belongs to L1(P ) , taking supremum over all t 2 D ,

we get k Xn" kD 2 L1(P ) . Therefore the family f k Xn kD ; Fn j �1 < n � n" g forms a

reversed submartingale, and in particular we have:

M := limn!�1E k Xn kD <1 .

And since j L(t) j = limn!�1E j Xn(t) j � limn!�1E k Xn kD =M for every t 2 D , then

k L kD <1 , or equivalently L 2 B(D) . Therefore it is no restriction to assume that the cover

= f D1; . . . ; Dm" g 2 �(D) satisfying (1) and (2) in (ii), also satisfies the following condition:

j L(t0)� L(t00) j < "

for all t0; t00 2 Dj and all j = 1; . . . ; m" . Hence for given t 2 Dj with 1 � j � m" , and

for every n � n" we may deduce:

j Xn(t)� L(t) j � j Xn(t)�Xn(dj) j + j Xn(dj)� L(dj) j + j L(dj)� L(t) j �� Ef jXn"(t)�Xn"(dj)j j Fn g+ j Xn(dj)� L(dj) j + " � Ef j j Fn g +

+ j Xn(dj)� L(dj) j + " � max1�j�m"

Ef j j Fn g+ max1�j�m"

j Xn(dj)� L(dj) j + " .

Taking supremum over all t 2 D we may conclude:

lim supn!�1

k Xn � L kD � lim supn!�1

max1�j�m"

Ef j j Fn g +

+ lim supn!�1

max1�j�m"

j Xn(dj)� L(dj) j + " = max1�j�m"

lim supn!�1

Ef j j Fn g +

+ max1�j�m"

lim supn!�1

j Xn(dj)� L(dj) j + " = max1�j�m"

Ef j j F�1 g+ " .

Hence by (2) in (ii) we easily obtain:

Pf lim supn!�1

k Xn � L kD > 2" g < "

for all " > 0 . Since L 2 B(D) , thus the family f k Xn�L kD ; Fn j �1 < n � n" g forms a

non-negative reversed submartingale which converges P -almost surely to zero, as n! �1 . But

then E k Xn�L kD ! 0 as n! �1 , and by (5) in Lemma 2.1 and Remark 3.3 we may conclude

that the family of reversed submartingales ( f j Xn(t)� L(t) j ; Fn j �1 < n � �1 g j t 2 T )is totally bounded in P -mean relative to D . This fact completes the proof.

20

Proposition 4.10




(i) Xn(t)! L(t) (a:s:) , as n ! �1 ; 8t 2 T

(ii) infn��1

E k Xn kD <1 .

If the family of reversed submartingales:


is totally bounded in P-mean relative to D , then we have:

(iii) 8" > 0 ; 9n" � �1 ; 9 = f D1; . . . ; Dm" g 2 �(D) and 91; . . . ;m" 2 L1(P )satisfying:

(1) j Xn"(t0)�Xn"(t

00) j � j ; 8t0; t00 2 Dj ; 8j = 1; . . . ; m"

(2) E( max1�j�m"

j) < " .

Proof. Let us first note that by (5) in Lemma 2.1 and Remark 3.3 the family of reversed

submartingales ( f j Xn(t)� L(t) j ; Fn j �1 < n � �1 g j t 2 T ) is totally bounded in P -

mean, if and only if the following condition is satisfied:

(1) limn!�1E k Xn � L kD = 0

Therefore under our assumptions the family f k Xn � L kD ; F j �1 < n � n0 g forms a

reversed submartingale, for some n0 � �1 . Since we obviously have:

k L kD � E k Xn � L kD + E k Xn kDfor all n � �1 , then by (ii) we get k L kD <1 , or equivalently L 2 B(D) . Now for a given

cover = f D1; . . . ; Dm g 2 �(D) let us define:

C;n = max1�j�m

supt0;t002Dj

j Xn(t0)�Xn(t

00) j

for all n � �1 . Then we have:

C;n � max1�j�m

supt0;t002Dj

( j Xn(t0) j + j Xn(t

00) j ) � 2 k Xn kD

for all n � �1 . Since by (ii) the family f k Xn kD ; Fn j �1 < n � n0 g forms a

reversed submartingale obviously satisfying infn�n0 E k Xn kD > �1 for n0 � �1 , then

f k Xn kD j n � n0 g is uniformly integrable, and therefore the family of random variables:

21

f C;n j C 2 �(D) ; n � n0 gis uniformly integrable. Thus for given " > 0 , there is 0 < � < "=2 satisfying:

(2) For every F 2 F satisfying P (F ) < � we have:ZFC;n dP < "=2

for all C 2 �(D) and all n � n0 .

And for such � > 0 , using the fact that L is bounded on D , we can find a finite cover

= f D1; . . . ; Dm" g 2 �(D) such that:

j L(t0)� L(t00) j < �=2

for all t0; t00 2 Dj and all j = 1; . . . ; m" . Let us note that:

j Xn(t0)�Xn(t

00) j � j Xn(t0)� L(t0) j + j L(t0)� L(t00) j + j Xn(t

00)� L(t00) j

for all t0; t00 2 Dj and all j = 1; . . . ; m" , so taking supremum over all t0; t00 2 Dj and

maximum over all j = 1; . . . ; m" we may deduce:

C;n = max1�j�m"

supt0;t002Dj

j Xn(t0)�Xn(t

00) j � 2 k Xn � L kD + �=2

Hence by (1) we have:

Pf C;n > � g � Pf k Xn � L kD > �=4 g < �

for all n � n" with some n" � �1 . Therefore by (2) we may conclude:

E(C;n) = E(C;n � 1f C;n�� g) + E(C;n � 1f C;n>� g) � � + "=2 < "

for all n � n" . The proof now follows easily by putting:

j = supt0;t002Dj

j Xn"(t0)�Xn"(t

00) j

for all j = 1; . . . ;m" .

Theorem 4.11




(i) Xn(t)! L(t) (a:s:) , as n ! �1 ; 8t 2 T

(ii) infn��1E k Xn kD <1 .

22

Then the following three statements are equivalent:

(1) 8" > 0 ; 9n" � �1 ; 9 = f D1; . . . ; Dm" g 2 �(D) and 91; . . . ;m" 2 L1(P )satisfying:

(i) j Xn"(t0)�Xn"(t

00) j � j ; 8t0; t00 2 Dj ; 8j = 1; . . . ; m"

(ii) Pf max1�j�m"

Ef j j F�1 g > " g < "

(2) 8" > 0 ; 9n" � �1 ; 9 = f D1; . . . ; Dm" g 2 �(D) and 91; . . . ;m" 2 L1(P )satisfying:


00) j � j ; 8t0; t00 2 Dj ; 8j = 1; . . . ; m"

(ii) E( max1�j�m"

j) < "

(3) The family of reversed submartingales:

(fj Xn(t)� L(t) j;Fn j �1 < n � �1g j t 2 T )


Proof. The implications (1)) (3) and (3)) (2) follow by proposition 4.9 and proposition

4.10, respectively. In order to prove the implication (2) ) (1) take " > 0 , then there exists

n" � �1 ; = f D1; . . . ; Dm" g 2 �(D) and 1; . . . ;m" 2 L1(P ) satisfying (i) in (2), and:

E( max1�j�m"

j) < "2 .

Hence by Markov’s inequality we may conclude:

Pf max1�j�m"

Ef j j F�1 g > " g � Pf Ef max1�j�m"

j j F�1 g > " g � 1

"E( max

1�j�m"

j) < " .

Thus (i) and (ii) in (1) are fulfilled, and the proof is complete.

Remark 4.12 Under the hypotheses in Theorem 4.11, suppose that the �-algebra F�1is degenerated, i.e. P (F ) 2 f0; 1g ; 8F 2 F�1 . Then Ef j j F�1 g = E(j) , for all

j = 1; . . . ; m" , and condition (ii) in (1) is equivalent to the following condition:

(1) max1�j�m"

E(j) � "

provided that " � 1 .

We turn to applications of the preceding results in the next example. In this paper we only

present those that concern the uniform law of large numbers in a straightforward way. For further

applications in statistics we shall refer the reader to [13].

23

Example 4.13

Let (S;A; �) be a probability space, let (;F ; P ) = (SN;AN; �N) be its countable product,

and let �n(!) = !n for ! = (!1; !2; . . . ) 2 and n � 1 . Then f �n j n � 1 g is a

sequence of independent identically distributed random functions defined on a probability space

(;F ; P ) with values in a measurable space (S;A) and with a common distribution law � .

Let T be a set, and let f : S � T ! R be an arbitrary function. Then one can find in [5] that

the following five statements are equivalent:

(1) The map f belongs to the space LLN0(B(T ); �) , or equivalently there exists m 2 B(T )such that:

k m� 1

n

nXj=1

f(�j) kT �! 0 (a:s:) and (P �) , as n ! 1

(2)

Z �k f(�1) kT dP <1 and there exists m 2 B(T ) such that:

k m� 1

n

nXj=1

f(�j) kT �! 0 (P �) , as n ! 1

(3) There exists m 2 B(T ) such that:

k m� 1

n

nXj=1

f(�j) kT �! 0 (a:s:)� , as n ! 1

(4) There exists m 2 B(T ) such that:

k m� 1

n

nXj=1

f(�j) kT �! 0 (L1)� , as n ! 1

(5) The map f is eventually totally bounded in �-mean, or equivalently the following three

conditions are satisfied:

(i) The map s 7! f(s; t) is �-measurable , 8t 2 T

(ii)

Z �k f(s) kT �(ds) <1

(iii) For each " > 0 , there exists = f D1; . . . ; Dm" g 2 �(T ) such that:

infn�1E

�( supt0;t002Dj

j 1n

nXi=1

f(t0; �i)� 1

n

nXi=1

f(t00; �i) j ) < " ; 8j = 1; . . . ; m" .

In this case we necessarily have:

(6) f( � ; t) 2 L1(�) , for all t 2 T

(7) If M(t) =

ZSf(s; t) �(ds) is the �-mean function associated to f , then M 2 B(T ) and:

k M � 1

n

nXj=1

f(�j) kT �! 0 (c) , as n ! 1

24

where (c) denotes either of the following convergence notions: (a:s:); (a:s:)�; (P �); (P�); (L1)�;(L1)� . Also to prove the implication (1)) (2) the following statement is established in [5]:

(8) If k m� 1

n

nXj=1

f(�j) kT �! 0 (a:s:) , as n!1 , for some m 2 B(T ) , thenZ �k f(�1) kT dP <1 .

Let us put X�n(t) = (1=n)Pn

j=1 f(�j; t) for all n � 1 and all t 2 T , and suppose that (6)

holds. Then ( f Xn(t) ; Fn j �1 < n � �1 g j t 2 T ) is a family of reversed martingales

indexed by T , where Fn is the permutation invariant �-algebra of order �n based on the

random function � = (�1; �2; . . . ) for n � �1 , see [1] . Consequently we can apply Theorem

4.1 and Theorem 4.3 together with Theorem 4.7 and Theorem 4.11, respectively, as well as their

consequences in Corollary 4.2 and Corollary 4.4. Furthermore, let us note that for the converse

statement in Theorem 4.3 it is enough to require L = M 2 B(T ) in order that:

k X�n � L kT = kM � 1

n

nXj=1

f(�j; t) kT �! 0 (a:s:) , as n ! 1

implies the validity of (ii) in Theorem 4.3 for each countable D subset of T . Indeed, in this

case we have:

k X�n � L kD = kM � 1

n

nXj=1

f(�j; t) kD �! 0 (P ) , as n ! 1

and by (8) we find:

infn�1E k X�n � L kD = inf

n�1E kM � 1

n

nXj=1

f(�j; t) kD �

� kM kT +

Z �k f(�1) kT dP <1 .

Hence the above statement follows by Theorem 3.1. Also note that our Corollary 4.4 extends

the implications among (1)-(5) to the general reversed martingale case. Moreover, by (8) and

Corollary 4.4 we may easily conclude, if L = M 2 C(T )\B(T ) , or T is hereditarily separable

and L 2 B(T ) , and if the following two conditions are satisfied:

k X�n � L kT = kM � 1

n

nXj=1

f(�j; t) kT �! 0 (a:s:) , as n ! 1

�(X�n)! 0 (c) , as n ! 1 ,

then we have:

k X�n � L kT = kM � 1

n

nXj=1

f(�j; t) kT �! 0 (c)

25

as n ! 1 , where (c) denotes either of the following convergence notions: (a:s:)�; (P �);(L1)�; (L1)� . In particular, if T is hereditarily separable, L 2 B(T ) , X�n(!; � ) 2 C(T ) for

a:s: ! 2 and all n � 1 , and:

k X�n � L kT = kM � 1

n

nXj=1

f(�j; t) kT �! 0 (a:s:) , as n!1 ,

then we have:

k X�n � L kT = kM � 1

n

nXj=1

f(�j; t) kT �! 0 (a:s:)� and (L1)�

as n!1 . Finally, let us note that condition (iii) in (5) can be reformulated in the following way:

(9) 8" > 0 ; 9n" � �1 ; 9 = f D1; . . . ; Dm" g 2 �(T ) and 1; . . . ;m" 2 L1(P ) sat-

isfying:


00) j � j ; 8t0; t00 2 Dj ; 8j = 1; . . . ;m"

(ii) max1�j�m"

E(j) < " .

Moreover, by the Hewitt-Savage 0-1 law the �-algebra F�1 =T1

n=1F�n is degenerated, see [1].

By using Remark 4.12 hence we could in fact observe that Theorem 4.11 examines the eventually

total boundedness in the mean condition for general families of reversed martingales, as well as

Theorem 4.7 and Remark 4.8 its reflection to families of reversed submartingales.

Example 4.14

Let (;F ; P ) = ([0; 1];B([0; 1]); �) , let T = [0; 1] with Euclidean topology, and let

f Zn ; n � 1 g be an arbitrary sequence of non-negative random elements defined on . Let

us define X�n(t; !) = Zn(!) � 1t(!) and X(t; !) = 0 , for all ! 2 , all t 2 T and all

n � 1 : Then the family:

f Xn(t) ; Fn j �1 < n � �1 g

forms a reversed martingale, for all t 2 T , where f Fn ; n � �1 g may be an arbitrary

increasing sequence of �-algebras on included in F . Clearly, we have:

k X�n(!)�X(!) kT = k (X�n(!)�X(!))+ kT =

= �(X�n(!)) = �+(X�n(!)) = Zn(!)

for all ! 2 and all n � 1 . Hence we may conclude that inequality (vi) given in Theorem

4.1 and Theorem 4.3 can not be improved in general.

Acknowledgment. The author would like to thank his supervisor, Professor J. Hoffmann-

Jørgensen, for instructive discussions and valuable comments.

26

REFERENCES

[1] CHOW, J. S. and TEICHER, H. (1978). Probability Theory. Springer-Verlag New York Inc.

[2] DOOB, J. (1953). Stochastic Processes. John Wiley and Sons, Inc.

[3] GINE, J. and ZINN, J. (1984). Some limit theorems for empirical processes. Ann. Probab.

12 (929-989).

[4] HARDY, G. H. (1917). On the convergence of certain multiple series. Proc. Cambridge

Phil. Soc. 19 (86-95).

[5] HOFFMANN-JØRGENSEN, J. (1984). Necessary and sufficient conditions for the uniform law

of large numbers. Proc. Probab. Banach Spaces V (Medford 1984), Lecture Notes in Math.

1153, Springer-Verlag Berlin Heidelberg (258-272).

[6] HOFFMANN-JØRGENSEN, J. (1985). The law of large numbers for non-measurable and non-

separable random elements. Asterisque 131 (299–356).

[7] HOFFMANN-JØRGENSEN, J. (1990). Uniform convergence of martingales. Proc. Probab.

Banach Spaces VII (Oberwolfach 1988), Progr. Probab. Vol. 21, Birkhauser Boston, (127-

137).

[8] MOORE, C. N. (1938). Summable Series and Convergence Factors. Amer. Math. Soc., New

York, Inc.

[9] MORICZ, F. (1979). On the convergence in a restricted sense of multiple series. Analysis

Math. 5 (135–147).

[10] NEVEU, J. (1975). Discrete Parameter Martingales. North Holland Publ Co. & American

Elsevier Publ. Co.

[11] PESKIR, G. (1991). Perfect measures and maps. Math. Inst. Aarhus, Preprint Ser. No. 26,

(34 pp).

[12] PESKIR, G. (1991). Vitali convergence theorem for upper integrals. Math. Inst. Aarhus,

Preprint Ser. No. 28, (14 pp). Proc. Funct. Anal. IV (Dubrovnik 1993), Various Publ. Ser.

Vol. 43, 1994 (191-203).

[13] PESKIR, G. (1992). Consistency of statistical models described by families of reversed sub-

martingales. Math. Inst. Aarhus, Preprint Ser. No. 9, (26 pp). Probab. Math. Statist. 18,

1998 (289-318).

[14] TALAGRAND, M. (1987). The Glivenko-Cantelli problem. Ann. Probab. 15 (837-870).

[15] VAPNIK, V. N. and CHERVONENKIS, A. Ya. (1981). Necessary and sufficient conditions for

the uniform convergence of means to their expectations. Theory Probab. Appl. 26 (532-553).

Goran Peskir

Department of Mathematical Sciences

University of Aarhus, Denmark

Ny Munkegade, DK-8000 Aarhus

home.imf.au.dk/goran

[email protected]

27

Uniform Convergence of Reversed Martingales€¦ · J. Theoret. Probab. Vol. 8, No. 2, 1995, (387-415) Preprint Ser. No. 21, 1992, Math. Inst. Aarhus Uniform Convergence of Reversed

Documents