advances in mathematical economics - 2013 - applications of birkhoff-kingman ergodic theorem.pdf

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Adv. Math. Econ. 16, 138 (2012)

Some applications of Birkhoff-Kingman

ergodic theorem

Charles Castaing1 and Marc Lavie2

1 Department de Mathematiques, Case courrier 051, Universite Montpellier II,

34095 Montpellier Cedex 5, France

(e-mail: [email protected])

2 Laboratoire de Mathematiques appliquees, Universite de Pau et des Pays de

L Adour, BP 1155, 64013, Pau cedex France

(e-mail: [email protected])

Received: June 30, 2011

Revised: September 27, 2011

JEL classification: C01, C02

Mathematics Subject Classification (2010): 28B20

Abstract. We present various convergence results for multivalued ergodic theorems

in Bochner-Gelfand-Pettis integration.

Key words: Conditional expectation, epiconvergence, ergodic, Bochner-Gelfand-

Pettis integration, Birkhoff-Kingman ergodic theorem, Mosco convergence, multi-

valued convergence, slice convergence

1. Introduction

Classical ergodic theorems for real valued random variables have been

recently extended into the context of epiconvergence in [7, 17, 24, 25, 34].

Using Abid result [1] on the a.s. convergence of subadditive superstationary

process, Krupa [27] and Schurger [32] treated the Ergodic theorems for sub-

additive superstationary families of convex compact random sets. Ghoussouband Steele [6] treated the a.s. norm convergence for subadditive process in

an order complete Banach lattice extending the Kingmans theorem for real

S. Kusuoka and T. Maruyama (eds.), Advances in Mathematical Economics 1

Volume 16, DOI: 10.1007/978-4-431-54114-1 1,

c Springer Japan 2012


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2 C. Castaing and M. Lavie

valued subadditive process. In this paper we present various applications of

the Birkhoff-Kingman ergodic theorem. The paper is organized as follows.In Sects. 34 we state and summarize for references some results on the

conditional expectation of closed convex valued integrable multifunctions in

separable Banach spaces and in their dual spaces. Main results are given in

Sects. 58. For the sake of completeness we provide an epiconvergence result

for parametric ergodic theorem in Sect. 5 that is a starting point of this study.

In Sect. 6 we treat the Mosco convergence for convex weakly compact valued

ergodic theorem in Bochner integration and the weak star Kuratowski con-

vergence for convex weakly star compact valued ergodic theorem in Gelfand

integration and also a scalar convergence result for convex weakly compactvalued ergodic theorem in Pettis integration. An unusual convergence for

superadditive integrable process in Banach lattice is given in Sect. 7 using

the integrable selection theorem for the sequential weak upper limit of a se-

quence of measurable closed convex valued random sets. Some relationships

with economic problems are also discussed. In Sect. 8 we present a conver-

gence theorem for convex weakly compact valued superadditive process in

Bochner integration via Komlos techniques.

2. Notations and preliminaries

Throughout this paper (, F, P ) is a complete probability space, (Fn)nN is

an increasing sequence of sub--algebras ofF such that F is the -algebra

generated by nNFn, E is a separable Banach space and E is its topological

dual. Let BE (resp. BE ) be the closed unit ball of E (resp. E) and 2E

the collection of all subsets of E. Let cc(E) (resp. cwk(E)) (resp. Lwk(E))

(resp. Rwk(E)) be the set of nonempty closed closed convex (resp. convex

weakly compact) (resp. closed convex weakly locally compact subsets of Ewhich contain no lines) (resp. ball-weakly compactclosed convex) subsets of

E, here a closed convex subset in E is ball-weakly compact if its intersection

with any closed ball in E is weakly compact. For A cc(E), the distance

and the support function associated with A are defined respectively by

d(x,A) = inf{x y : y A}, (x E)

(x, A) = sup{x, y : y A}, (x E).

We also define

|A| = sup{||x|| : x A}.

A sequence (Kn)nN in cwk(E) scalarly converges to K cwk(E) if

limn (x, Kn) =

(x, K), x E. Let B be a closed bounded


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Some applications of Birkhoff-Kingman ergodic theorem 3

convex subset ofE and let C be the closed convex subset ofE. Then the gap

[4] D(B,C) between B and C is denoted by

D(B,C) = inf{||x y|| : x B, y C}.

By Hahn Banach theorem we know that

D(B,C) = supxBE

{(x, C) (x, B)}.

Given a sub--algebra B in , a mapping X : 2E is B-measurable if

for every open set U in E the set

XU := { : X() U = }

is a member ofB. A function f : E is a B-measurable selection of X

iff() X() for all . A Castaing representation of X is a sequence

(fn)nN ofB-measurable selections ofX such that

X() = cl{fn(), n N}

where the closure is taken with respect to the topology of associated with

the norm in E. It is known that a nonempty closed-valued multifunction X : c(E) is B-measurable iff it admits a Castaing representation. IfB is

complete, the B-measurability is equivalent to the measurability in the sense

of graph, namely the graph ofX is a member ofBB(E), hereB(E) denotes

the Borel tribe on E. A cc(E)-valued B-measurable X : cc(E) is

integrable if the set S1X(B) of all B-measurable and integrable selections of

X is nonempty. We denote by L1E (B) the space of E-valued B-measurable

and Bochner-integrable functions defined on and L1cwk(E)

(B) the space of

all B-measurable multifunctions X : cwk(E) such that |X| L1R(B).

We refer to [16] for the theory of Measurable Multifunctions and ConvexAnalysis, and to [18, 29] for Real Analysis and Probability.

3. Multivalued conditional expectation theorem

Given a sub--algebra BofF and an integrable F-measurable cc(E)-valued

multifunction X : E, Hiai and Umegaki [20] showed the existence of a

B-measurable cc(E)-valued integrable multifunction denoted by EBX such

thatS

1EBX

(B) = cl{EBf : f S1X(F)}

the closure being taken in L1E (, A, P ); EBX is the multivalued conditional

expectation ofX relative to B. IfX L1cwk(E)(F) and the strong dual Eb is


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separable, then EBX L1cwk(E)(B) with S1EBX

(B) = {EBf : f S1X (F)}.

A unified approach for general conditional expectation of cc(E)-valued in-tegrable multifunctions is given in [33] allowing to recover both the cc(E)-

valued conditional expectation of cc(E)-valued integrable multifunctions in

the sense of [20] and the cwk(E)-valued conditional expectation ofcwk(E)-

valued integrably bounded multifunctions given in [5]. For more informa-

tion on multivalued conditional expectation and related subjects we refer to

[2, 9, 16, 20, 24, 33]. In the context of this paper we summarize a specific

version of conditional expectation in a separable Banach space.

Proposition 3.1. Assume that the strong dual E

b

is separable. LetBbe a sub-

-algebra ofF and an integrableF-measurable cc(E)-valued multifunction

X : E. Assume further there is a F-measurable ball-weakly compact

cc(E)-valued multifunction K : E such that X() K() for all

. Then there is a unique (for the equality a.s.) B-measurable cc(E)-

valued multifunction Y satisfying the property

() v LE (B),

(v(), Y ())dP () =

(v(), X())dP ().

EBX := Y is the conditional expectation ofX.

Proof. The proof is an adaptation of the one of Theorem VIII.35 in [16]. Let

u0 be an integrable selection ofX. For every n N, let

Xn() = X() (u0() + nBE ) n N .

As X() K() for all , we get

Xn()=X()(u0()+nBE)K()(u0()+nBE ) n N .

As K() is ball-weakly compact, it is immediate that Xn L1cwk(E)(F).

so that, by virtue of ([5] or ([33], Remarks of Theorem 3), the conditional

expectation EBXn L1cwk(E)

(B). It follows that

()

(v(), EBXn())P(d) =

(v(), Xn())P(d)

n N, v LE (B). Now let

Y() = cl(nNEBXn()) .


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Then Y is B-measurable and a.s. convex. Now the required property ()

follows from () and the monotone convergence theorem. Indeed

n N, v LE (B), u0, v (v,Xn)

(v,X)

v, EBu0 (v,EBXn)

(v,Y).

Now the uniqueness follows exactly as in the proof of Theorem VIII.35 in

[16], via the measurable projection theorem ([16], Theorem III.32).

4. Conditional expectation in a dual space

Let (, F, P ) be a complete probability space, (Fn)nN an increasing

sequence of sub -algebras of F such that F is the -algebra generated

by n1Fn. Let E be a separable Banach space, D1 = (xp)pN is a dense

sequence in the closed unit ball of E, E is the topological dual of E,

BE (resp. BE ) is the closed unit ball of E (resp. E). We denote by Es

(resp. Ec ) (resp. Eb ) the topological dual E

endowed with the topology

(E, E) of pointwise convergence, alias w topology (resp. the topology

c of compact convergence) (resp. the topology s associated with the dualnorm ||.||Eb

), and by Em the topological dual E endowed with the topology

m = (E, H ), where H is the linear space ofE generated by D, that is the

Hausdorff locally convex topology defined by the sequence of semi-norms

pk(x) = max{|x, xp| : p k}, x

E, k 1.

Recall that the topology m is metrizable by the metric

dEm

(x1 , x2 ) :=

p=p=1

1

2p|xp, x1 xp, x2 |, x1 , x2 E.

Further we have

dEm

(x, y) ||x y||Eb ,x, y E E.

We assume from now that dEm

is held fixed. Further, we have m w

c s. When E is infinite dimensional these inclusions are strict. On the

other hand, the restrictions ofm

, w

, to any bounded subset ofE

coincideand the Borel tribes B(Es ), B(Ec ) and B(E

m ) associated with E

s , E

c and

Em are equal. Noting that E is the countable union of closed balls, we

deduce that the space Es is Suslin, as well as the metrizable topological space

Em . A 2Es -valued multifunction (alias mapping for short) X : Es is


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F-measurable if its graph belongs to F B(Es ). Given a F-measurable

mapping X : Es and a Borel set G B(Es ), the set

XG = { : X() G = }

is F-measurable, that is XG F. In view of the completeness hypothesis

on the probability space, this is a consequence of the Projection Theorem (see

e.g. Theorem III.23 of [16]) and the equality

XG = proj {Gr(X) ( G)}.

Further if u : Es is a scalarly F-measurable mapping, that is, for

every x E, the scalar function x,u() is F-measurable, then the

function f : (,x) ||x u()||Eb is F B(Es )-measurable, and for

every fixed , f(,.) is lower semicontinuous on Es , shortly, f is a

normal integrand, indeed, we have

||x u()||Eb= sup

jN

ej, x u()

here D1 = (ej)j1 is a dense sequence in the closed unit ball ofE. As eachfunction (,x) ej, x

u() isFB(Es )-measurable and continuous

on Es for each , it follows that f is a normal integrand. Consequently,

the graph ofu belongs to FB(Es ). Besides these facts, let us mention that

the function dEb (x, y) = ||xy||Eb

is lower semicontinuous on Es Es ,

being the supremum of w-continuous functions. If X is a F-measurable

mapping, the distance function dEb (x, X()) is F-measurable, by us-

ing the lower semicontinuity of the function dEb (x, .) on Es and measurable

projection theorem ([16], Theorem III.23) and recalling that Es is a Suslin

space. A mapping u : E

s is said to be scalarly integrable, if, for everyx E, the scalar function x, u() is F-measurable and integrable.

We denote by L1E [E](F) the subspace of allF-measurable mappings u such

that the function |u| : ||u()||Eb is integrable. The measurability of|u|

follows easily from the above considerations. By cwk(Es ) we denote the

set of all nonempty convex (E, E)-compact subsets of Es . A cwk(Es )-

valued mapping X : Es is scalarly F-measurable if the function

(x, X()) is F-measurable for every x E. Let us recall that any

scalarly F-measurable cwk(Es )-valued mapping is F-measurable. Indeed,

let (ek)kN be a sequence in E which separates the points ofE, then we have

x X() iff, ek , x (ek, X()) for all k N. By L1cwk(Es )(,F, P )

(shortlyL1cwk(Es )(F)) we denote the of all scalarly integrable cwk(E)-valued

multifunctions X such that the function |X| : |X()| is integrable, here

|X()| := supyX() ||y||Eb , by the above consideration, it is easy to see


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that |X| is F-measurable. For the convenience of the reader we recall and

summarize the existence and uniqueness of the conditional expectation inL1cwk(Es )

(F). See ([33], Theorem 3).

Theorem 4.1. Given a L1cwk(Es )(F) and a sub--algebra BofF, there

exists a unique (for equality a.s.) mapping := EB L1cwk(Es )(B), that is

the conditional expectation of with respect to B, which enjoys the following

properties:

a)

(v,)dP =

(v, )dP for all v LE (B).

b) E

B

||BE

a.s.c) S1 (B) is sequentially (L1E

[E](B), LE (B)) compact (here S1 (B) de-

notes the set of all L1E [E](B) selections of) and satisfies the inclusion

EBS1 (F) S1 (B).

d) Furthermore one has

(v,EBS1(F)) = (v, S1 (B))

for all v LE (B).

e) EB is increasing: 1 2 a.s. implies EB1 EB2 a.s.

This result involves the existence of conditional expectation for (E, E)

closed convex integrable mapping in E, namely

Theorem 4.2. Given a F-measurable (E, E) closed convex mapping

in E which admits a integrable selection u0 L1E [E](F) and a sub--

algebra BofF. For every n N and for every let

n() = () (u0() + nBE ).

() = (E, E) cl[EBn()].

Then () is a.s. convex and is a B-measurable (E, E) closed convex

mapping which satisfies the properties:

a)

(v, )dP =

(v,)dP for all v LE (B).

b) := EB is the unique for equality a.s. B-measurable (E, E) closed

convex mapping with property a).

c) EB is increasing: 1 2 a.s. implies EB1 E

B2 a.s.

By definition, := EB is the conditional expectation of.

Proof. Follows the same line of the proof of Theorem VIII-35 in [16] and is

omitted.

Before going further we state and summarize an epiconvergence result

[6] which is a starting point of our study.


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5. An epiconvergence result

The following is a special version of an epiconvergence result in [6]. We

summarize and give the details of proof because it has an own interest and

leads to the convergence of multivalued Birkhoffs ergodic theorem.

Proposition 5.1. Let E be a separable Banach space, T a measurable

transformation of preserving P, I the -algebra of invariant sets and

f : Es R be a F B(Es )-measurable normal convex integrand

such thatf+(, 0) = 0 for all and letf the polar off. Assume that

f

(, u())dP () < u L

BE (, I, P ) .

The following epiconvergence result holds, for any u LBE

(, I, P ), one

has:

() supkN

lim supn

infyBE

[1

n

n1j=0

f(Tj ,y ) + k||u() y||]

g(, u()) = EIf(, u()) a.s.

where

g(,x) = ((x, 1), EIepif (, .)) (,x) E.

and

() supkN

lim infn

infyBE

[1

n

n1j=0

f(Tj ,y ) + k||u() y||]

g(, u()) = EIf(, u()) a.s.

Proof. Step 1 Here we use some arguments developed in the proof of Propo-

sition 5.5 in [7]. Let

() := epif (, .) .

Then is a closed convex integrable mapping in Es R and admits anintegrable selection, namely

(0, 0) () Es R .

Let EB be the conditional expectation of whose existence is given by

Proposition 4.2. Then the conditional expectation g of the normal integrand

f is a I B(E)-measurable normal integrand satisfying

A

f(, v())dP () =

A

g(, v())dP ()


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for all v LE (,I, P ), and for all A Iwith

f(, v()) := ((v(), 1), epif (, .))

g(, v()) := ((v(), 1), EIepif (, .)) .

Hence g(, v()) = EIf(, v()) is positive and integrable for all v

LBE

(, I, P ). Let us set

(,x) := g(,x) (,x) BE.

and

k (,x) = infyBE

[(,y) + k||x y||] (,x) BE .

Then

supkN

k(,x) = (,x) (,x) BE .

Let u LBE

(,I, P ). Let p N. Since E is separable Banach space,

applying measurable selection theorem ([16], Theorem III-22) yields a

(I,B(E))-measurable mapping vk,p,u : BE such that

0 (, vk,p,u()) + k||u() vk,p,u()|| k(, u()) +

1

p .

From the classical Birkhoff ergodic theorem (see e.g. [34]) we provide a neg-

ligible set Nk,p,u such that for / Nk,p,u

limn

1

n

n1j=0

f(Tj, vk,p,u()) = EI[f(,vk,p,u())]

= (,vk,p,u()) a.s.

Whence we provide a negligible set Nk,p,u such that

lim supn

infyBE

[1

n

n1j=0

f(Tj ,y ) + k||u() y||E]

lim supn

[1

n

n1j=0

f(Tj, vk,p,u()) + k||u() vk,p,u()||E]

= (, vk,p,u()) + k||u() vk,p,u()||E k(, u())

+1

p / Nk,p,u.


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Taking the supremum on k N in this inequality yields a negligible set

Nu := kN,pNNk,p,u such that for / Nu

supkN

lim supn

infyBE

[1

n

n1j=0

f(Tj ,y ) + k||u() y||E]

supkN

k(, u()) = (, u()) = g(, u()).

Step 2 For k N, n N and for (,x) BE , we set

(,x) := f(,x)

k(,x) = infyBE

[f(,y) + k||x y||E].

n(,x) =1

n

n1j=0

f(Tj,x) =1

n

n1j=0

(Tj,x).

kn(,x) = inf

yBE

[n(,y) + k||x y||E].

Then the following hold

(5.1.1) |k(,x) k(,y)| k||x y||E (,x,y) BE BE .

(5.1.2) 0 k(,x) (,x) (,x) BE .

(5.1.3) sup

kN

k(,x) = (,x) (,x) BE .

There is a negligible set N which does not depend on x BE such that

(5.1.4) EI(,x) =

supkN E

Ik(,x) if \ N BE0 if (,x) N BE

where EIk and EI denote the conditional expectation relative to I of

k and respectively. Further we have

(5.1.5) kn (,x) 1

n

n1j=0

k(Tj,x) (,x) BE .


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By virtue of classical Birkhoff ergodic theorem (see e.g. Lemma 5 in [34]) it

follows that

(5.1.6) limn

1

n

n1j=0

k(Tj, u()) = EI[k(, u())] a.s.

(5.1.5) -(5.1.6) yield a negligible set Nk,u such that

lim infn

infyBE

[1

n

n1

j=0

(Tj ,y ) + k||u() y||E]

lim infn

[1

n

n1j=0

k(Tj, u())]

= EI[k(, u())] / Nk,u .

Using (5.1.4) and the preceding convergences, we produce a negligible set

Nu = kNNk,u N such that

supkN

lim infn

infyBE

[ 1n

n1j=0

(Tj,y) + k||u() y||E]

EI(, u()) = g(, u()) / Nu.

For more on epiconvergence results in ergodicity one may consult

[6, 7, 17, 24, 34].

6. Birkhoff ergodic theorem for Bochner-Gelfand-Pettisintegrable multifunctions

We need to recall some classical results on the slice convergence. For more

information on the slice topology one may consult [3, 4].

Lemma 6.1. Assume that E is strongly separable. Let D1 be a dense se-

quence in the closed unit ball BE ofE. Then for all bounded closed convex

subsets B andC in X, the following holds:

D(B,C) : = supxBE

{(x, C) (x, B)}

= supxD1

{(x, C) (x, B)}.


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Proof. Equality

D(B,C) = supxBE

{(x, C) (x, B)}

follows from Hahn-Banach theorem, while the second equality

supxBE

{(x, C) (x, B)} = supxD1

{(x, C) (x, B)}

follows from the strong separability of E, noting that the function x

(x

, C)

(x

, B) is strongly continuous on E

.

Lemma 6.2. Assume thatE is separable. LetD1 be a dense sequence in the

closed unit ball BE ofE with respect to the Mackey topology. Then for all

convex weakly compact subsets B andC in E, the following holds:

D(B,C) : = supxBE

{(x, C) (x, B)}

= supxD1

{(x, C) (x, B)}.

Proof. Equality

D(B,C) = supxBE

{(x, C) (x, B)}

follows from Hahn-Banach theorem, while equality

supxBE

{(x, C) (x, B)} = supxD1

{(x, C) (x, B)}

follows, noting that the function x (x, C) (x, B) is contin-

uous on E with respect to the Mackey topology.

We begin to treat the Mosco convergence for multivalued ergodic theorem

in Bochner integration.

Theorem 6.1. Assume that E and its strong dual Eb are separable. Let T

be a measurable transformation of preserving P, I the -algebra of in-

variant sets andX : cwk(E) a F-measurable and integrably bounded

mapping, i.e. X L1cwk(E)(, F, P ), then the following hold:

() limn

(x,1

n

n1j=0

X(Tj)) = (x, EIX())


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for all x BE and almost surely , and

() limn

d(x,1

n

n1j=0

X(Tj)) = d(x,EIX())

for all x E and almost surely , consequently

() limn

D(B,1

n

n1j=0

X(Tj)) = D(B,EIX())

for all closed bounded convex subsets B ofE and almost surely . Here

EIX is the conditional expectation ofX in the sense of Hiai-Umegaki.

Proof. Let D1 = (ek )kN be a dense sequence in the closed unit ball BE

with respect to the the norm dual topology. Note that the mapping |X|

|X|() = supxBE

|(x, X())|

is F-measurable and integrable (see e.g. [16]). Next, applying the classicalBirkhoff ergodic theorem to |X| and each (ek , X) yields almost surely

(6.1.1) limn

1

n

n1j=0

|X|(Tj) = EI|X|().

limn

(ek ,1

n

n1

j=0

X(Tj)) = limn

1

n

n1

j=0

(ek ,X(Tj))

(6.1.2) = (ek , EIX()).

Claim (1):

lim infn

D(B,1

n

n1j=0

X(Tj)) D(B,EIX())

for all closed bounded convex subsets B E and almost surely .

Indeed, for each k N, there is a negligible set Nk such that (6.1.2) holds.

Then N := k Nk is negligible and for \ N we have


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lim infn D(B,

1

n

n1j=0

X(Tj

))

= lim infn

supxBE

[(x,1

n

n1j=0

X(Tj)) (x, B)|

supxD1

lim infn

[(x,1

n

n1j=0

X(Tj)) (x, B)]

= supxD1

[ lim supn

(x, 1n

n1j=0

X(Tj)) (x, B)]

= supxD1

[(x, EIX()) (x, B)] = D(B,EIX()).

Claim (2):

EIX() s-li1

n

n1j=0

X(Tj) a .s.

Recall thatS1

EIX= cl{EIf : f S1X}

here the closure is taken in the sense of the norm in L1E (see also Theorem 3

in [33]). Let f S1X. Applying Birkhoff ergodic theorem for Banach valued

functions (see [26], Theorem 2.1, p.167) yields

limn

1

n

n1j=0

f (Tj) = EIf () a .s.

Whence for a.s. , EIf () is the strong limit of 1n

n1j=0 f (T

j)

Sn() :=1n

n1j=0 X(T

j). In other words

EIf () s-li1

n

n1j=0

X(Tj) a.s.

Now if g S1

EI

X

, there is a sequence (fn) in S1X such that E

Ifn g a.s.

so that

g() s-li1

n

n1j=0

X(Tj) a .s.


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Using this inclusion and taking a Castaing representation of EIX we can

assert that1

EIX() s-li1

n

n1j=0

X(Tj) a .s.

Claim (3):

(6.1.3)

limn

(x,1

n

n1j=0

X(Tj)) = (x, EIX()) a.s. x BE .

(6.1.4) limn

d(x,1

n

n1j=0

X(Tj)) = d(x,EIX()) a.s x E.

There is a negligible set N0 such that for each \ N0

supnN

1

n

n1j=0

|X|(Tj) <

and there is a negligible set Nk such that for each \ Nk

limn

(ek,1

n

n1j=0

X(Tj)) = (ek, EIX())

and also there is a negligible set M such that for each \ M

EIX() s-li1

n

n1j=0

X(Tj).

Then N = k0Nk M is negligible. Let \N, e D1 and x

BE .

We have the estimate

|(x,1

n

n1j=0

X(Tj))(x, EIX())|||xe|| supnN

1

n

n1j=0

|X|(Tj)

+|(e,1

n

n1j=0

X(Tj)) (e, EIX())|

+||x

e

||EI

|X|().

1 This argument is classical, see the proof of Theorem 4.6 in [ 8].


16/38


Then from the preceding estimate, it is immediate to see that for \ N

and x BE we have

limn

(x,1

n

n1j=0

X(Tj)) = (x, EIX())

which proves (6.1.3). Let \ N we have

EIX() s-li1

n

n1

j=0

X(Tj)

so that

lim supn

d(x,1

n

n1j=0

X(Tj)) d(x,EIX()) x E.

Further for (,x) \ N E we have

(6.1.5) lim infn d(x,

1

n

n1j=0 X(T

j

))

= lim infn

supxB

E

[x, x (x,1

n

n1j=0

X(Tj))]

supkN

limn

[ek , x (ek ,

1

n

n1j=0

X(Tj))]

= supkN

[ek , x (ek , EI

X())]

= d(x,EIX()).

Note that (6.1.5) follows also from Claim (1) by taking B = {x}. Hence for

(,x) \ N E we have

limn

d(x,1

n

n1j=0

X(Tj)) = d(x,EIX())

which proves (6.1.4). Formally using (6.1.4) we get

(6.1.6) lim supn

D(B,1

n

n1j=0

X(Tj)) D(B,EIX()) a.s.


17/38


for all closed bounded convex set B E because we have

lim supn

D(B,1

n

n1j=0

X(Tj)) = lim supn

infxB

d(x,1

n

n1j=0

X(Tj))

infxB

lim supn

d(x,1

n

n1j=0

X(Tj))

= infxB

d(x,EIX()) = D(B,EIX()).

So combining (6.1.6) with Claim (1) we get

limn

D(B,1

n

n1j=0

X(Tj)) = D(B,EIX()) a.s.

for all closed bounded convex set B ofE.

Remark. Since X L1cwk(E)

(,F, P ), and the strong dual Eb is separable,

by [33], EBX L1cwk(E)

(I) with S1EIX

(I) = {EBf : f S1X(F)}. Using

this special property of the EIX, it is easy to prove in inclusion

w-ls1

n

n1j=0

X(Tj) EIX() a.s.

Let x w-ls Sn() with \ N, there is xp x weakly for some

xp Snp () (p 1). As

ek , x = limpek , xp limp

(ek , Snp ()) = (ek , E

IX()) k N

we get x EIX() because EIX() is convex weakly compact (see [16],Proposition III.35). From this inclusion and Claim (2) we have

Mosco - limn

1

n

n1j=0

X(Tj) = EIX() a.s.

It is worth to address the question of validity of Theorem 6.1 when the

strong dual is no longer separable. For this purpose we will provide a vari-

ant of Theorem 6.1 by using some compactness assumptions. Nevertheless

this need a careful look. A convex weakly compact valued measurable map-ping X : cwk(E) is weak compactly integrably bounded if there

exist L1R+

(, F, P ) and a convex weakly compact set K in E such that

X() ()K for all . It is obvious that weak compactly integrably

bounded implies integrably bounded.


18/38


Theorem 6.2. Assume thatE is separable, T is a measurable transformation

of preserving P,I is the -algebra of invariant sets andX : cwk(E)is a F-measurable and weak compactly integrably bounded mapping, then

the following hold:

limn

(x,1

n

n1j=0

X(Tj)) = (x, EIX())

for all x BE and almost surely , and

limn

d(x,1

n

n1j=0

X(Tj)) = d(x,EIX())

for all x E and almost surely , and consequently

limn

D(B,1

n

n1j=0

X(Tj)) = D(B,EIX())

for all convex weakly compact subsets B ofE and almost surely . HereEIX is the conditional expectation ofX in the sense of Hiai-Umegaki.

Proof. By assumption there exist L1R+

(, F, P ) and a convex weakly

compact set K in E such that X() ()K for all . W.l.o.g we

may assume that K is equilibrated. It is not difficult to see that EIX()

EI()K for all . Hence EIX() is convex weakly compact. Let

D1 = (ek )kN a dense sequence in the closed unit ball BE with respect to

the Mackey topology. Next, applying the classical Birkhoff ergodic theorem

to and each

(e

k , X) yields

(6.2.1) limn

1

n

n1j=0

(Tj) = EI() a.s.

limn

(ek ,1

n

n1j=0

X(Tj))= limn

1

n

n1j=0

(ek ,X(Tj))

(6.2.2) = (ek , EIX()) a.s.


19/38


Consequently by (6.2.1) 1n

n1j=0 (T

j) is pointwise bounded a.s., say

() := supnN

1

n

n1j=0

(Tj) < a.s.

It follows that

Sn() :=1

n

n1j=0

X(Tj) [1

n

n1j=0

(Tj)]K ()K a.s.

Claim (1):

lim infn

D(B,1

n

n1j=0

X(Tj)) D(B,EIX())

for all convex weakly compact subset B E and almost surely .

Indeed, for each k N, there is a negligible set Nk such that (6.2.2) holds.

Then N := k Nk is negligible and for \ N we have

lim infn

D(B,1

n

n1j=0

X(Tj))

= lim infn

supxBE

[(x,1

n

n1j=0

X(Tj)) (x, B)|

sup

x

D

1

lim infn

[(x,1

n

n1

j=0

X(Tj)) (x, B)]

= supxD1

[ lim supn

(x,1

n

n1j=0

X(Tj)) (x, B)]

= supxD1

[(x, EIX()) (x, B)] = D(B,EIX()).

Claim (2):

E

I

X() s-li

1

n

n1

j=0

X(T

j

) a .s.

Recall that

S1EIX

= cl{EIf : f S1X}


20/38


here the closure is taken in the sense of the norm in L1E (see also Theorem 3

in [33]). Firstly we check that

S1EIX

S1s-li 1n

n1j=0 X(T

j)

Let f S1X. Applying Birkhoff ergodic theorem for Banach valued functions

(see [26], Theorem 2.1, p.167) yields

limn

1

n

n1

j=0

f (Tj) = EIf () a .s.

Whence for a.s. , EIf () is the strong limit of 1n

n1j=0 f (T

j)

Sn() :=1n

n1j=0 X(T

j). In other words

EIf () s-li1

n

n1j=0

X(Tj) a.s.

Now if g S1

EIX. There is a sequence (fn) in S1

X such that EI

fn g a.s.so that

g() s-li1

n

n1j=0

X(Tj) a .s.

Using this inclusion and taking a Castaing representation of EIX we can

assert that

EIX() s-li1

n

n1j=0

X(Tj) a .s.

Now we are going to prove

Claim (3):

(6.2.3)

limn

(x,1

n

n1j=0

X(Tj)) = (x, EIX()) a.s. x BE .

(6.2.4) limn

d(x,1

n

n1j=0

X(Tj)) = d(x,EIX()) a.s x E.


21/38


There is a negligible set N0 such that for each \ N0

() := supnN

1

n

n1j=0

(Tj) <


limn

(ek,1

n

n1j=0

X(Tj)) = (ek, EIX())

and also there is a negligible set M such that for each \ M

EIX() s-li1

n

n1j=0

X(Tj).

Then N = k0Nk M is negligible. Let \ N, x BE and > 0

Pickej D1 such that

max{

(x

ej, E

I

()K),

(ej x

, EI

()K)} <

and

max{(x ej,()K),(ej x

, ()K)} < .

Let us write

|(x, Sn()) (x, EIX())|

= |(x, Sn()) (ej, Sn())|

+|(e

j

, Sn()) (e

j

, EIX())|

+|(ej, EIX()) (x, EIX())|.

As Sn() ()K and EIX() EI()K for all n N and for all

\ N, we have the estimates

|(x, Sn()) (ej, Sn())|

max{(x ej,()K),(ej x

,()K)} <

and

|(ej, EIX()) (x, EIX())|

max{(x ej, EI()K), (ej x

, EI()K)} < .


22/38


Finally we get

[(x, Sn())(x, EIX())] < [(ej, Sn())

(ej, EIX())]+2.

As [(ej, Sn()) (ej, E

IX())] 0, from the preceding estimate, it

is immediate to see that for \ N and x BE we have

limn

(x,1

n

n1j=0

X(Tj)) = (x, EIX())

which proves (6.2.3). Let \ N we have

EIX()) s-li1

n

n1j=0

X(Tj)

so that

lim supn

d(x,1

n

n1

j=0

X(Tj)) d(x,EIX()) x E.

Further for (,x) \ N E we have

(6.2.5) lim infn d(x,1n

n1j=0

X(Tj))

= lim infn

supxB

E

[x, x (x,1

n

n1j=0

X(Tj))]

supkN

limn

[ek , x (ek ,

1

n

n1j=0

X(Tj))]

= supkN

[ek , x (ek , E

IX())]

= d(x,EIX()).

Note that (6.2.5) follows also from Claim (1) by taking B = {x}. Hence for

(,x) \ N E we have

limn

d(x,1

n

n1j=0

X(Tj)) = d(x,EIX())


23/38


which proves (6.2.4). Formally using (6.2.4) we get

(6.2.6) lim supn

D(B,1

n

n1j=0

X(Tj)) D(B,EIX()) a.s.

for all closed bounded convex set B E because we have

lim supn

D(B,1

n

n1j=0

X(Tj)) = lim supn

infxB

d(x,1

n

n1j=0

X(Tj))

infxB

lim supn

d(x,1

n

n1j=0

X(Tj))

= infxB

d(x,EIX()) = D(B,EIX()).

So combining (6.2.6) with Claim (1) we get

limn

D(B,1

n

n1

j=0

X(Tj)) = D(B,EIX()) a.s.

for all convex weakly compact set B ofE.

For more information on the conditional expectation of multifunctions,

we refer to [2, 9, 20, 33]. In particular, recent existence results for condi-

tional expectation in Gelfand and Pettis integration as well as the multival-

ued Dunford-Pettis representation theorem are available [2, 9]. These results

involve several new convergence problems, for instance, the Mosco conver-

gence of sub-super martingales, pramarts in Bochner, Pettis or Gelfand inte-

gration, see [2, 9, 10, 12].The following variant deals with Gelfand integration, here E is no longer

strongly separable. A mapping : cwk(Es ) is Gelfand-integrable, if

the mapping (e, (.)) is integrable, for all e E. The Aumann-Gelfand

integral of, denoted by

AG-

GdP = {G-

f dP : f S }

where S is the set of all Gelfand-integrable selections of and G- f dP

is the Gelfand integral of f S . We need to recall the following result on

the existence of conditional expectation of convex weak star compact valued

Gelfand-integrable mapping ([9], Theorem 6.1).

Theorem 6.3. LetB be a sub--algebra ofF and let X be a cwk(Es )-

valued Gelfand-integrable mapping such thatEB|X| [0, +[. Then there


24/38


exists a uniqueB-measurable, cwk(Es )-valued Gelfand-integrable mapping,

denoted by Ge-EBX which enjoys the following property: For every h L(B), one has

() AG-

hGe-EBXdP = AG-

hXdP.

Ge-EBX is called the Gelfand conditional expectation of X.

We need the following definition.

Definition 6.1. The Banach space E is weakly compact generated (WCG) if

there exists a weakly compact subset ofE whose linear span is dense in E.

Theorem 6.4. Let E be a separable WCG Banach space, T a measurable

transformation of preserving P, I the -algebra of invariant sets andX :

cwk(E) a F-measurable and Gelfand-integrable mapping such that

EI|X| [0, +[. Then the following hold:

w-Kuratowski - limn

1

n

n1j=0

X(Tj) = EIX() a.s.

where EIX := Ge-EIX (for short) denotes the Gelfand conditional expec-

tation ofX.

Proof. Let D1 = (ek)kN be a dense sequence in BE. Note that the mapping

|X|

|X|() = supxBE

|(x, X())|

is F-measurabie (see e.g. [16] or [2]). Next, applying the classical real valued

Birkhoff ergodic theorem to |X| and each (ek , X) yields

(6.4.1) limn

1

n

n1

j=0

|X|(Tj) = EI|X|() a.s.

limn

(ek,1

n

n1j=0

X(Tj)) = limn

1

n

n1j=0

(ek,X(Tj))

(6.4.2) = (ek, EIX()) a.s.

Consequently there is a negligible set N0 such that for each \ N0

(6.4.3) supnN

1

n

n1j=0

|X|(Tj) <


25/38



(6.4.4) limn

(ek,1

n

n1j=0

X(Tj)) = (ek, EIX()).

Claim

w-Kuratowski - limn

1

n

n1j=0

X(Tj) = EIX() a.s.

Set N = k0Nk. Then N is negligible. Let \ N, e D1 and x BE .

We have the estimate

|(x,1

n

n1j=0

X(Tj)) (x,EIX())| ||x e|| supnN

1

n

n1j=0

|X|(Tj)

+|(e,1

n

n1j=0

X(Tj)) (e,EIX())|

+||x e||EI|X|().

Then from (6.4.3) and (6.4.4) and the preceding estimate, it is immediate tosee that for \ N and x BE we have

(6.4.5) limn

(x,1

n

n1j=0

X(Tj)) = (x,EIX()).

Now since E is WCG, from (6.4.3)-(6.4.4)-(6.4.5) and ([19], Theorem 4.1)

we deduce that

w-Kuratowski - limn

1

n

n1

j=0

X(Tj) = EIX() a.s.

Now we treat the multivalued ergodic theorem in Pettis integration. A

mapping : cwk(E) is Pettis-integrable, if the mapping (e, (.))

is integrable for all e E and if any scalarly integrable selection of is

Pettis-integrable. The Aumann-Pettis integral of is defined by

AP-

dP = {P-

fdP : f SPe }

where SPe is the set of all Pettis-integrable selections of and P-

fdP

is the Pettis integral of f SP e . We need to recall the following result on

the existence of conditional expectation of convex weak star compact valued

Pettis-integrable mapping ([2], Theorem 4.4).


26/38


Theorem 6.5. Assume thatEb is separable. LetBbe a sub--algebra ofF

and letX be a cwk(E)-valued Pettis-integrable mapping such thatEB|X| [0, +[. Then there exists a unique B-measurable, cwk(E)-valued Pettis-

integrable multifunction, denoted by P e-EBX, which enjoys the following

property: For every h L(B), one has

P e-

hP e-EBXdP = P e-

hXdP

where P e-

hP e-EBXdP and P e-

hXdP denote the cwk(E)-valued

Aumann-Pettis integral of hP e-EB

X andhX respectively.

Theorem 6.6. Assume that E and its strong dual Eb are separable, T is

a measurable transformation of preserving P, I is the -algebra of in-

variant sets andX : cwk(E) is a F-measurable and Pettis-integrable

mapping such thatEI|X| [0, +[.

Then the following hold:

limn

(x,1

n

n1

j=0

X(Tj)) = (x, EIX())

for all x BE and almost surely , where EIX := P e-EIX (for

short) denotes the Pettis conditional expectation ofX.

Proof. Let D1 = (ek )kN a dense sequence in BE with respect to the norm

dual topology. Note that the mapping |X|

|X|() = supxBE

|(x, X())|

isF-measurabile (see e.g. [16]). Next, applying the classical Birkhoff ergodic

theorem to |X| and each (ek , X) (see e.g. [34]) yields almost surely

(6.6.1) limn

1

n

n1j=0

|X|(Tj) = EI|X|() a.s.

limn (ek ,

1

n

n1j=0

X(Tj)) = limn

1

n

n1j=0

(ek ,X(Tj))

(6.6.2) = (ek , EIX()).


27/38


Consequently there is a negligible set N0 such that for each \ N0

(6.6.3) R() := supnN

1

n

n1j=0

|X|(Tj) <


(6.6.4) limn

(ek ,1

n

n1j=0

X(Tj)) = (ek , EIX())

Set N = k0Nk . Then N is negligible. Let \ N, x BE and

e D1 . By (6.6.3) we have

|Sn()| R() <

for each \ N. We have the estimate

|(x, ,1

n

n1

j=0

X(Tj)) (x, EIX())| ||x e||R()

+|(e,1

n

n1j=0

X(Tj)) (e, EIX())|

+||x e||EI|X|().

Then from (6.6.3) and (6.6.4) and the preceding estimate, it is immediate to

see that for \ N and x BE we have

limn

(x,1

n

n1j=0

X(Tj)) = (x, EIX()).

7. An unsual convergence for superadditive integrable

process in Banach lattice

We must recall some facts from ergodic theory and provide the definition of

superadditive process in L1R(,F, P ). A sequence (Sm)mN in L1R(,F, P )

is a superadditive process provided that, for all m, n N, we have

Sm+n() Sm() + Sn(Tm)

for all .


28/38


Theorem (J.F.C. Kingmann) [18]. For any superadditive sequence (Sn)nN

of integrable random variable, ( Snn )nN converges a.s. as n to

:= supnNEISn

n. Here is integrable if and only if supnN

E(Sn)n

< ,

and if is integrable, then (Snn

)nN converges to also in L1.

In the following let E denotes a separable Banach lattice and E+ denotes

the positive cone in the dual E. For background on properties of Banach

lattices, we may consult Schafer [31], Peressini [30]. The following definition

is formally similar to the case of real valued integrable superadditive process.

Definition 7.1. Let E be a separable Banach lattice, let T a measurabletransformation of preserving P. A sequence (Sm)mN in L

1E (,F, P ) is

a superadditive process provided that for all m, n N, and for all ,

we have

Sm+n() Sm() + Sn(Tm)

for all .

The above superadditive condition means that for all f E+, for all

m, n N, for all , we have

f, Sm+n() f, Sm() + Sn(T

m).

In otherwords, for each f E+, (f, Sm)mN is a superadditive process

in L1R(, F, P ).

The following is an unusual convergence of superadditive vector valued

process with localization of the limit. Surprisingly this result can be con-

sidered as a by product of some results in Mathematical Economics dealing

with the existence of integrable selection of the sequential weak upper limit

of a sequence of measurable multifunctions. See Theorem 4.9 in [12]. At

this point, let us mention some important variants. Krupa [27] and Schurger

[32] treated the Ergodic theorems for subadditive superstationary families of

convex compact random sets using Abid result [1] on the convergence of sub-

additive superstationary process while Ghoussoub and Steele [6] treated the

a.s. norm convergence for subadditive process in an order complete Banach

lattice extending the Kingmans theorem for real valued subadditive process.

Theorem 7.1. Assume thatE is separable Banach lattice, E+ is the positive

cone in the dual E, T is a measurable transformation of preserving P, I

is the -algebra of invariant sets and (Sn)nN is a superadditive process inL1E (, F, P ) satisfying:

(i) There is a sequence (rn)nN in L0R(,F, P ) with rn co{|

Sii

| : i n}

such thatlim sup rn L1R(,F, P ).


29/38


(ii) For each f E, (f,Sn

n)nN is uniformly integrable.

(iii) There is a convex ball weakly compact valued multifunction : Esuch that

Sn()

n () n N .

Then there exists Z L1E (, F, P ) verifying:

(a)

Af, ZdP = limn

A

f, Snn

dP =

AsupnNf

,EISn

ndP

for all A F and for all f E+.

(b) Z() m1 cl co{Sn()

n: n m} almost surely.

Proof. Let f E+. It is clear that each (f, Sn)nN is a real valued inte-

grable superadditive process. By condition (ii) and Kingmann ergodic theo-

rem, f, Snn

converges a.s. and also in L1 to the integrable function

supnN

EIf, Sn

n= sup

nN

f,EISn

n

which entails

limn

f,

A

Sn

ndP = lim

n

A

f, Sn

ndP (7.1)

(7.1.1) =

A

supnN

f,EISn

ndP A F.

Now using (i)(ii)(iii) and repeating the arguments of the proof of Theo-

rem 4.9 in [12] we provide a sequence (Zn)nN whose members are convex

combinations of( Snn

) and Z L1E (,F, P ) such that

(7.1.2) weak- limn

Zn() = Z() a.s.

Using (7.1.2) we get (b)

Z()

m1

cl co{Sn()

n: n m} a.s.

By (ii) we note that the sequence (f, Zn)nN is uniformly integrable, for

each f

E

, which entails

(7.1.3)

A

f, ZdP = limn

A

f, ZndP


30/38


for all A F. Now let f E+. Coming back to (7.1.1)- (7.1.2)-(7.1.3)

we get A

f, ZdP = limn

A

f, ZndP = limn

f,

A

ZndP

= limn

f,

A

Sn

ndP = lim

n

A

f,Sn

ndP =

A

supnN

f,EISn

ndP .

thus proving (a).

Here is a corollary of Theorem 7.1.

Corollary 7.1. Assume that E is reflexive separable Banach lattice, T is a

measurable transformation ofpreserving P,I is the -algebra of invariant

sets and(Sn)nN is a superadditive process in L1E (, F, P ) satisfying:

(i) supnNE(|Sn|)

n< .

(ii) For each f E, (f,Sn

n)nN is uniformly integrable.

Then there exists Z L1E (, F, P ) verifying:

(a)

Af

, ZdP = limn

Af

,

Sn

n dP =

A supnNf

,

EISn

n dPfor all A F and for all f E+.

(b) Z()

m1 cl co{Sn()

n: n m} almost surely.

Proof. We give an alternative proof. By (i) one has

supnN

e,Sn

ndP <

for all e E. Then by Kingmann ergodic theorem, for each f E+,

f

,Snn converges a.s. and also in L

1

to the integrable function

supnN

EIf, Sn

n= sup

nN

f,EISn

n

which entails

limn

f,

A

Sn

ndP = lim

n

A

f,Sn

ndP =

A

supnN

f,EISn

ndP A F

Since by (i) the sequence ( Snn )nN is bounded in L1E (,F, P ) and E isreflexive, there is a sequence (Zn)nN whose members are convex combina-

tions of( Snn

) and Z L1E (,F, P ) such that

(7.2.1) limn

Zn() = Z() a.s.


31/38


(see e.g. [11], Proposition 6.7), that is (Zn)nN norm converges a.s. to Z.

Now we can finish the proof as in Theorem 7.1.

Remarks. Theorem 7.1 can be considered as a by product of a convergence

result related to the Fatou lemma in Mathematical Economics. See ([12],

Theorem 4.9). That is a surprise establishing the link between convergence in

superadditive process in Probability and Fatou Lemma in Mathematical Eco-

nomics. Theorem 7.1 is an extension of Kingman ergodic theorem to vector

valued integrable superadditive sequences. In both Theorem 7.1 and its corol-

lary, it is easily seen that the sequence ( Snn

)nN is Mazur-tight in the sense of

[12]. The use of Mazur trick is decisive here.

8. Convex weakly compact valued superadditive process

in L1cwk(E)

(,F, P )

In this section we will develop some Komlos techniques in the convergence

of convex weakly compact valued superadditive process via the Kingman er-

godic theorem for superadditive process in L1R(, F, P ). We need to provide

the definition of superadditive process in L1cwk(E)(, F, P ).

Definition 8.1. A sequence (Sm)mN in L1cwk(E)(, F, P ) is a superadditive

process provided that for all m, n N, for all , we have

Sm() + Sn(Tm) Sm+n().

Further, the superadditivity given in this definition is equivalent to

(x, Sm()) + (x, Sn(T

m)) (x, Sm+n()) x E.

so that coming back to the definition of real valued integrable superadditive

process, we see that, for each x E, the sequence ((x, Sm))mN is a

superadditive integrable process in L1R(, F, P ). When equality is assumed

to hold in the definition 8.1, (Sm)mN is called additive. Compare with [28].

In the remainder of this paper, E is a separable Hilbert space. The following

lemma is a useful and has its own interest.

Lemma 8.1. LetE be a separable Hilbert space. Let (Xn)nN be a bounded

sequence in L1cwk(E)(, F, P ). Then there is a subsequence (X(n))nN of

(Xn)nN andX in L1cwk(E)(, F, P ) such that

limn

(x,1

n

ni=1

X(i)()) = (x,X()) a.s. x BE.


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limn d(x,

1

n

n

i=1

X(i)()) = d(x,X()) a.s. x E.

Proof. Step 1 Let (ek)kN be a dense sequence in BE . For each fixed k N

and n N, pick an integrable selection uk,n ofXn such that

ek, uk,n() = (ek, Xn()) .

As the sequence (|Xn|)nN is bounded in L1R(,F, P ) and the sequence

(uk,n )nN is bounded in L1E (, F, P ) for each k N, by applying Komlos

theorem2

in Hilbert space [23] and a diagonal procedure we find an applica-tion : N N, L1R(, F, P ), uk, L

1E (,F, P ) satisfying

(8.1.1) limn

1

n

ni=1

|X(i)|() = () a.s.

and for each k N

(8.1.2) limn

1

n

n

i=1

uk,(i)() = uk,() a.s.

with respect to the norm ofE.

Step 2 Let us set

Yn() :=1

n

ni=1

X(i)()

so that by (8.1.1) supnN |Yn| < almost surely; and

X() = s-li Yn()

so that for every k N

uk,() X() a.s .

Then for every k N we have

limn

(ek, Yn()) = limn

1

n

ni=1

ek , uk,(i)

= ek , uk,() (ek, X() a.s.

2 A more general version of Komlos theorem for B convex reflexive Banach space is

available in [21].


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and

(ek, X()) limn

(ek, Yn()) a.s.

so that

(ek, X()) = limn

(ek, Yn()) a.s.

Using a density argument as in the previous results in section 6 we deduce

that

(8.1.2) (x,X()) = limn

(x,Yn()) a.s. x E.

Finally X is measurable and satisfies the inequality

|X|dP

lim infn

|Yn|dP supn

|Xn|dP < .

Step 3 Now we claim that

limn

d(x,Yn()) = d(x,X()) a.s. x BE .

From (8.1.2) deduce that

(8.1.3) lim infn d(x,Yn())

= lim infn

supyBE

[y, x (y,Yn())]

supk

limn

[ek, x (ek , Yn())]

= supk

[ek, x (ek, X())]

= d(x,X()).

for all x E and almost surely . By definition ofX we have

(8.1.4) lim supn

d(x,Yn()) d(x,X())

for all x E and all . So the claim follows from (8.1.3)-(8.1.4).

Now we provide our convergence theorem for convex weakly compact

valued superadditive process in Hilbert spaces.

Theorem 8.1. Assume thatE is separable Hilbert space, T is a measurable

transformation of preserving P, I is the -algebra of invariant sets and

(Sn)nN is a superadditive process in L1cwk(E)

(, F, P ) satisfying


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(i) ( Snn

)nN is uniformly integrable.

(ii) There is L Lwk(E) such that Snn () L, n N, .

Then there are a subsequence (Z(n))nN of (Snn

)nN and Z L1cwk(E)

(, F, P ) satisfying the following properties:

limn

(x,1

n

ni=1

Z(i)()) = (x,Z()) a.s. x BE .

limn

d(x,1

n

n

i=1

Z(i)()) = d(x,Z()) a.s. x E.

Consequently

A

(x,Z)dP = limn

A

(x,Sn

n)dP =

A

supnN

(x,EISn)

ndP

for each x BE and each A Fwith the localization property

Z() co w-lsSn()

n

a.s.

Proof. Let x BE. Then by our assumption and by virtue of Kingman er-

godic theorem, the uniformly integrable real valued superadditive sequence

(((x, Snn

))nN converges a.s. and in L1 to the integrable function

x := supnN

EI(x,Sn)

n= sup

nN

(x,EISn)

n

which entails

(8.2.1) limn

A

(x,Sn

n)dP =

A

supnN

(x,EISn)

ndP

for each A F. Since ( Snn

)nN is bounded in L1cwk(E)(, F, P ), by

Lemma 8.1 we find a subsequence (Z(n))nN of (Snn

)nN and Z

L1cwk(E)(, F, P ) satisfying

(8.2.2) lim

n

(x,1

n

n

i=1

Z(i)()) = (x,Z()) a.s. x BE .

(8.2.3) limn

d(x,1

n

ni=1

Z(i)()) = d(x,Z()) a.s. x E.


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From (8.2.1)-(8.2.2) and Lebesgue-Vitali theorem, it follows that

limn

A

(x,Sn()

n)dP() = lim

n

A

(x,1

n

ni=1

Z(i)())dP()

(8.2.4) =

A

(x,Z())dP()

for every x E and for every A F. Now we claim that

(8.2.5) Z() co w-ls Sn()n

a.s.

As Sn()n

is included in L Lwk(E), the Lwk(E)-valued mapping

co w-lsSn()

nis measurable, see e.g. ([12], Theorem 5.4). Assume that (8.2.5)

does not hold. Using Lemma III. 34 in [16] provides x E and a F-

measurable set A with P (A) > 0 such that

(x,Z()) > (x,cow-ls

Sn()

n)

for all A. For each n N pick a maximal integrable selection zn ofSnn

associated with x, i.e.

x, zn() = (x,

Sn()

n) .

Then it is obvious that the sequence (zn)nN is relatively sequentially com-

pact in L1E (, F, P ), see e.g. ([13], Theorem 6.2.5 or Theorem 6.4.13).

So we may assume that (zn)nN (L1E , L

E ) converges to a function z

L1E (, F, P ) with the localization property ([13], Proposition 6.5.3)

z() co w-ls zn() co w-lsSn()

na.s.

As zn z weakly in L1E (, F, P ) we get

limn

A

x, zndP =

A

x, zdP .

By combining with (8.2.4) it follows thatA

x, z()dP() =

A

(x,Z())dP ().


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As z() co w-lsSn()

na.s. on A, by integrating on A we deduce that

A

(x,cow-lsSn()

n)dP()

A

(x,Z())dP()

that contradicts the inequalityA

(x,cow-lsSn()

n)dP()

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