Convergence of two-parameter banach space valued martingales and the radon-nikodym property of banach spaces

Acta Mathematica Sinica, English Series 1999, April, Vol.15, No.2, p. 187-196

Acta Mathematica Sinica, English Series 9 Springer-Verlag1999

A. S. Convergence of Two-Parameter Banach Space Valued Martingales and the Radon-Nikodym Property of Banach Spaces Shixin Gan Department of Mathematics, Wuhan University, Wuhan 430072, P. R. China

A b s t r a c t In this paper, we prove that under the F4 condition, any L log + L bounded two-parameter Banach space valued martingale converges almost surely to an integrable Banach space valued random variable if and only if the Banach space has the Radon-Nikodym property. We further prove that the above conclusion remains true if the F4 condition is replaced by the weaker local F4 condition. Keywords Two-parameter Banach space valued martingale, A. S. convergence, Radon-Nikodym property, F4 condition, Local F4 condition 1 9 9 1 M R S u b j e c t Classification 60Bll, 60G46, 60G60 Let (fl, .T, P ) be a complete probability space and (B, I1"II) a B a n a c h space. A strong measurable function defined on (f~, 5r, P ) with values in a B a n a c h space B is called a B valued r a n d o m variable. In this paper by measurability and convergence we m e a n strong measurability and strong convergence. A n expectation and a conditional expectation are the expectation and the conditional expectation in Bochner's integral sense. Let us denote the set of nonnegative integers by IV-+. N~_ = {z = (rn, n ) : m , n E N+}. For z = ( m , n ) , z ' = ( m ' , n ' ) E N~_, set z < z' i f m < rn~,n < n ~. We write z v z' = ( m V m ' , n V n ' ) ,

zAz' = (mAre',nAn').

Let {9r . . . . (m, n) E N~_} be a family of increasing sub-o--algebras of .T. We always suppose t h a t Jr0,0 contains all null sets. For any z = (rn, n) E N~. we introduce the following a-algebras:

Received June 20, 1997, Accepted January 8, 1998 Project supported by the National Natural Science Foundation of China and the State Education Commission Ph.D. Station Foundation

Shixin Gan

188

Cairoli and Walsh [1] introduced the concept of the F4 condition for a family {~'~, z 9 N~_} of increasing sub-a-algebras of ~-. It is known that under the F4 condition, every L log + L bounded two-parameter real valued martingale converges almost surely. It is not difficult to construct a uniformly bounded(naturaly L log + L bounded) two-parameter Banach space valued martingale {Xm,,~,.Tm,,~,(m,n) e N~_} where {Xm,n) doesn't converge a.s. and {Srm,,~} satisfies the F4 condition. Now our question is: W h a t are the sufficient and necessary conditions for a.s. convergence of all two-parameter Banach space valued martingales? In this paper we prove that under the F4 condition (or the local F4 condition) every L log + L bounded two-parameter Banach space valued martingale converges almost surely if and only if the Banach space has the Radon-Nikodym property RNP.

1

F4 C o n d i t i o n and L o c a l F4 C o n d i t i o n

D e f i n i t i o n 1.1 A family {J:z, z 9 N 2 } of increasing sub-a-algebras of jz is said to satisfy the F4 condition if for each z 9 N~, jzl, ~2 are conditionally independent with respect to jz , i e. I o r each z 9

VA 9

VB 9

then

P(AB[~ ) = P(AI~)P(B[~

).

The two-parameter martingale theory and the general theory of two-parameter stochastic processes are established on the basis of the F4 condition. Many fundamental theorems and results in these theories do not hold if the F4 condition is not satisfied. Therefore in recent years some authors have done much research work to relax the F4 condition. Among them Xingwu Zhuang and Jitao Li [2] introduced the following local F4 condition instead of the F4 condition. D e f i n i t i o n 1.2

Let ~, G1, ~2 be sub-a-algebras of J:. G E ~. If VAi 6 ~i, i = 1, 2, we have P(A1A2 [ G)IG = P ( A t I ~)P(A2 [ ~ ) I c ,

then ~1 and ~2 are called conditionally independent with respect to G on G. If G1 and G2 are conditionally independent with respect to G1 N ~2 on G, then G1 and ~2 are called conditionally independent on G. The following two lemmas are not difficult to prove. So their proofs are omitted. L e m m a 1.1 Let F = {jz . . . . (m, n) 6 N ~ } be a family of increasing sub-a-algebras of jr. Then the following statements are equivalent: (i) F satisfies the F4 condition, (ii) For any z = (rn, n) E N~, any integrable B valued random variable X , then E(E(XI71

)IT

z) =

E(E(XI = )I71z) =

(iii) For any integrable B valued random variable X and Vz, z' 6 N~, then

E(E(XIIz)IIz,) = E(X[Iz^=,). L e m m a 1.2 equivalent:

Let ~1,G2 be sub-a-algebras of JZ,G E ~1 n ~2. The following statements are

A. S. Convergence of Two-Parameter Banach Space Valued Martingales and RNP of Banach Spaces

189

(i) ~1 and ~2 are conditionally independent on G, (ii) For any bounded B valued random variable X , then E ( E ( X [ g~) ] g2)./G = E ( E ( X [ g2) [ gl)IG : E ( X [ gl CI g2)IG, (iii) For any X E L 1 (f~, gl, P; B), then E ( X [ g2)Ia E L 1 (f~, gl r3 g2, P; B ) , symmetrically, for any X E L 1 (ft, G2, P; B), then E ( X [ GI)Ia E L ~ (~2, gl n g2, P ; B ) . D e f i n i t i o n 1.3 Let M be a sub-direction of N~, G E ~eM'7:~" {:TZz,z E M } is said to satisfy the F4 condition on G if for any z, z' C M, j r and ~ , are conditionally independent with respect to J:~^z, on G. { j r , z E N~ } is said to satisfy the local F4 condition if there exists a sequence of sets {Gn} with Gi M Gj = 0, (i 7s j) and Un Gn = f~, and for each n, 3z~ E N 2 such that Gn E J=~. and { j r z > zn} satisfies the F4 condition on Gn. Obviously if F = {~'~, z E N~_} satisfies the F4 condition, then F must satisfy the local F4 condition. But the converse may not be true (see [2]). Therefore the local F4 condition is weaker than the F4 condition. We need the following lemma (see [2, Theorem 2.10]). L a m i n a 1.3 {F~, z E N~_} satisfies the local F4 condition if and only if there exist sequences Gn ~ f~ and zn ~ such that for each n, Gn E Jrzn and {Jr~,z > z~} satisfies the F4 condition on an.

B V a l u e d M a r t i n g a l e C h a r a c t e r i z a t i o n o f t h e F4 C o n d i t i o n a n d t h e Local F4 Condition A Banach space valued process X = {Xz, 9rz, z E N~.} is said to be adapted to {Srz, z E N~_} iff Xz is 5r~ measurable for every z E N_~. A Banach space valued process X = {Z~, 5r~, z E N~_} is said to be 1-adapted iff Xz is ~-1 measurable. Similarly we can define a 2-adapted process. Let X = {Xz, 5rz, z E N~_} be an integrable B valued adapted process. X is said to be a martingale if Vz, z' E N~., z _< z', we have E(X~, I .T~)

=

X~.

X is said to be a 1-martingale if for every fixed n E IV+, {Xm,n, 9r l , m E N+} is a one-parameter B valued martingale. A 2-martingale is defined in a similar way. T h e o r e m 2.1 Let F = {jz z E N 2 } be a family of increasing sub-~-algebras of J=. Then the following statements are equivalent: (i) F satisfies the F4 condition, (ii) Vz, z' E N ~ , V X E L 1 (ft, Srz,P;B), then E ( X I .~z,) E L 1 ( f ~ , ~ A ~ , , P ; B ) , forVz, z E (iii) / f {Z~, ~ , z E N~_} is a B valued martingale, then E(X

I

=

Shixin Gan

190

(iv) For any integrable B valued random variable X, setting Xz : E ( X I ~-~),Vz E N~_, then { Xz, Jvz, z E N~ } is both a 1-martingale and a 2-martingale, (v) Every B valued martingale {Xz, jc , z E N 2} is both a 1-martingale and 2-martingale.

Proof ( i ) ~ ( i i )

Vz, z' E N ~ , V X E Lx(f~,J=r~,P;B), it follows from L e m m a 1.1 t h a t E ( X I .T'~,) =

E(E(XI.,VJ].~,)

= E ( X I ~z^~,),

i.e. E ( X I J=~,) E L 1 ( f l , h r ~ ^ ~ , , P ; B ) . (ii)=~(iii) Let {Zz,J:~,z E N~_} be any B valued martingale. Vz, z' E N~, it follows from (ii) t h a t E(X~ I 9r~,) = E(E(X~ I Jr~,)]Jr, A~,)

=

E(X~ I.~^~,).

(iii)=~(iv) Let X be any integrable B valued r a n d o m variable. Set X~ = E ( X ] J : J , V z E N~_. Obviously {X~,JCz,z E N~_} is a B valued martingale. For fixed n E N+,m,m' E N + , m < m ' , any n' E N+, n < n', we have

E(Xm,,n I ~ m , n ' ) -- X . . . . Therefore

/ xm,,n@=f

VAe U n'>n

It follows by the m o n o t o n e class t h e o r e m t h a t

Clearly X,~,~ is .T~ measurable. Hence E ( X m , ~ 1~-~) = X . . . . i.e. {Xz, 5rz, z E N 2} is a B valued 1-martingale. Similarly we can prove t h a t it is a B valued 2-martingale. (iv)=~(i) For any z E N~_, any b o u n d e d B valued r a n d o m variable X , setting X~ = E ( X ] ~ ) , Vz E N~, {X~,Jr~,z E N 2} is a B valued martingale. It follows from (iv) t h a t {X~,JCz,z E N~} is b o t h a 1-martingale and a 2-martingale. Plainly for fixed n E N + , {E(XliTm,,~),J: . . . . m E N + } is a one p a r a m e t e r B valued martingale. T h u s

E(XI'~"~"~)-+E( xlvmeN+Ym"0

a's' (L1)"

For any m ' , m E N + , m < m', it follows from the 1-martingale p r o p e r t y t h a t

E ( E ( X [ ~ m ' , n ) [ ~ ) = E ( X m ' , ~ l g r ~ ) = Xm,n = E(XI$-m,~). L e t t i n g m ' --+ co, we obtain by the d o m i n a t e d convergence t h e o r e m of the conditional expectation that E(E(X[,~2)[:~) = E ( Z i ~ r m , ~ ) . In a similar way we can prove

E(E(XI~)I.~) E(Xl~'m,~). :

A. S. Convergence of Two-Parameter Banach Space Valued Martingales and R N P of Banach Spaces

191

Now let X be any integrable B valued r a n d o m variable. T h e n there exists a B valued simple function sequence {Xn}such t h a t ]IX,~]] < 2HXII(n > 1) and Z n --+ X a.s. From the above proof we have

E(E(Xn[.T'I~)[.T "2) = E(E(Znlt'~)[Y'~) = E(XnIY'~)

Vz e N~_.

Letting n -+ oo, it follows from the d o m i n a t e d convergence t h e o r e m of the conditional expectation t h a t

E(E(XI~'~)IY2~) = E(E(Xl.~)l.r~z) =

E(XI~).

It follows from L e m m a 1.1 t h a t F satisfies the F4 condition. ( v ) ~ ( i v ) is obvious. Now ( i ) ~ ( v ) . Let X = {Xz,gv~, z E N~_} be any B valued martingale. Clearly X~ is b o t h 5v~ and .~2 measurable. For each fixed n E N+, Vm, m' E N + , m < m ' and Vn' > n, it follows from L e m m a 1.1 t h a t

E(Xm',niY:m,n') = E(E(X,vnlYrm,,,~)]J:m,n) = E(Zm,,,~IYrm,n) = X . . . . Hence for any n' _> n, any E E 5~,~,n, we obtain

f x,,,',n@ = fzxm,n@. So n'>n

A p p l y i n g the m o n o t o n e class t h e o r e m it follows t h a t

fEXm',ndP= fExm,ndP,

VE E a

Therfore

E ( X ~ , , n ] -T'~) : X . . . . This shows t h a t X is a 1-martingale. martingale.

B y a similar a r g u m e n t we can prove t h a t X is a 2-

T h e o r e m 2.2 Assume zo E N~ and G E Yr~o. Then the following statements are equivalant: (i) {gv~, z _> Zo} satisfies the F4 condition on G, (ii) For any z, z' >_ zo and any bounded B valued random variable X , then

E( E( X[.T'~ )[.T'~, )IG = E( E( X[.T'z, )[.T'~)IG = E( X[.T'zAz, )IG, (iii) For any z >_ zo, and any bounded B valued random variable X , then

E(E(X[.T'I~)].T'~)IG = E(E(X[.T'2~)[.Tq~)IG = E(X[.T'z)IG, (iv) For any z, z' > zo and any X E L 1 (~2, j r , p; B), then

E(X[.%,)Ic ~

LI(~2, :TZ~n~,, P; B),

(v) For any B valued martingale {X~, J:~, z E N~}, if z, z' > Zo, then

192

Shixin Gan

(vi) For any bounded B valued random variable X , set X , = E(XIJ=z)IG. Then { X~. .T~[G. z > z0} is both 1-martingale and 2-martingale, (vii) For any B valued martingale { X z , ~ z , z E N 2 } , { X z I o , ~ , i G , z > Zo} is both 1martingale and 2-martingale.

This theorem can be proved by an argument similar to that in Theorem 2.1 by using Lemma 1.2. Details of the proof are omitted.

A. S. C o n v e r g e n c e of T w o - P a r a m e t e r B Valued M a r t i n g a l e s a n d R N P of B a n a c h Spaces In this section we always suppose that a family F = {~-,, z E N~_} of increasing sub-a-algebras of ~- satisfies the F4 condition unless otherwise stated. L a m i n a 3.1 Let {Xz,J:~,z E N~} be a B valued martingale. (i) For any A > O, then AP ( s u p ,,X.[I > A) _< - - e ( s u p EiiX,=I]log+ilX.I[+ 1 ) ; e- 1 \~N~ \zEN~_ (ii) For any p > 1, then E(sup

Proof If

<

[]X~[[v)

sup zEN~.

EIIx~II".

sup~EN~_(EIIX~Itlog + llX~ll) =

~ , then (i) holds. We suppose that sup,eN~(E[[Zz][ log + [[Xz[[) < oo. It follows from Theorem 2.1 that for each fixed m E N+, {X . . . . ~,~,n 2 E N+} is a one-parameter B valued martingale. Hence {l[Z,~,n[[,~2,n E N+} is a non-negative submartingale. Similarly for each n E N+, {[[Zm,,~l[, 5"~, m E N+} is a non-negative submartingale. Let Ym = supneN+ [[Xm,,~][. Obviously, sup

re,hEN+

IIX~,d[

=

sup mEN+

Kin.

It follows from the Doob inequality that EYm = E sup IIX~,dl hEN+

~ --

e

e -- 1

sup EIIXm,,~ll log + IIX~,,~ll+ 1)


hEN+

Clearly Ym is ~-~ measurable. For any m, rn' E N+, m < m', then

/ E(Ym, I5"~) = E ! sup llx

,,oll I

\hEN+

>_ sup /

(irx ,, ll i j: 1 ) _> sup ,x

hEN+

,oll =

hEN+

i.e. {Ym,-7"1, m E N+ } is a non-negative submartingale. By the maximal inequality, it follows that AP(sup mEN+

Ym>A)<

sup E Y m < -

-

mEN+

--

e

sup EIIX~,.II log + IlXm.II + 1).

-

i

rn,nEN+

A. S. Convergence of Two-Parameter Banach Space Valued Martingales and RNP of Banach Spaces

193

(ii) We can prove (ii) in a similar manner by the Doob inequality (p > 1).

Let B be a Banach space with Radon-Nikodym property and { X , , ~ , z E N 2 } a B valued martingale satisfying

T h e o r e m 3.1

sup EIIXzll log + IIXzll < oo zEN~_

(this condition is called the condition of L l o g + L boundedness). Then there exists an integrable B valued random variable X ~ such that lim X~ = X ~

zEN~_

a.s.

Proof First suppose that SUpzEN~_ EIIX=II v < ~ ( p > 1). In view of Corollary 4 and Corollary 2 of [3, p. 126] there exists an L p integrable B valued random variable Xoo such that L p

Xm,n --+ Xoo and

Xm,n = E ( X ~ l ~m,~),

Vm, n e N+.

So we can choose a sequence (ink, nk) E N 2, k > 1, (rnk, nk) "~ (oo, oo) such that oo

}--~ kPEIIZm~,n~ - X o o l F < oo. k=l

For each fixed k, consider the following two-parameter B valued martingale

{Xm,n --Xm~,,~,]: . . . . m k m k , n k nk}. It follows from L e m m a 3.1 that

E sup EIIX~,,~- x~,,~ll~ ~ Ap sup EIIXm,n - Xm~,,~ IIp ~ rn>rna .>n k

where Ap,

B p E I I X ~ - Xm~,n~ I[p,

m>m k n.n k

Bp are

constants only depending on p. Therefore oo

E P k=l

sup EIIX~,,~ - X ~ , n ~ II > rn>m

k

k=l

n_>nk

By the Borel-Cantelli lemma we obtain

1)

sup ]lXm,,~ - X~a,,~ k ][ < ;

i=i k=i

~>-~ ">nk

L p

= 1.

This shows that {Xm,~} is a.s. convergent. Since Xm,n -+ X ~ , so Xm,n --+ X ~ a.s. Now suppose that supm,,~ey+ E[[Xm,n[[ log + []Zm,,~[[ < oc. Obviously {Xm,,~} is uniformly integrable and supra,neW + E[[Xm,,~[] < oo. It follows from Corollary 4 and Corollary 2 of [3, p. 126] that there exists an integrable B valued r a n d o m varable X ~ such that L 1

Xm,,~ -+ X ~

Shixin Gan

194

and X~,,~ = E ( X ~ I Y-~,~),

Vm, n, E N+.

For any a > 0, let X(~ ) = Z ~ I ( J I Z ~ l l _< a) and X m(~) , n = E(X(~ ) ] 9vm,~). Clearly {Zm,~ X rl%rl, (a) j1- is a two-parameter B valued martingale. It follows from Lemma 3.1 that

\rn,nEN+

e ( sup EIIXm,,~- X(a,)I{ log+ I{X,~,,~- X(~,)II + 1) -< A ( e - 1) \,~,=eN+

_< ~(e--7~_ 1 (EIIX~ - X(~)ll log + IIX~ - X(~)ll + 1) <(x~.

Hence for any e > 0 we can choose sequences ek $ 0 and 0 < ak j" c~ such that P

sup

IlX~,,~-X(~,)ll_>ek

_< ~-r

for all

k_> 1.

\re,hEN+

Set ~ rn,nE N+

k=l

On E ~, for each k we have sup

IIXm,n - X(ma)~]l ~ ek.

rn,nE N+

Consequently l i m ~ know

X (~)m,,~= Xm,,~ uniformly with respect to m, n. From the above proof we a.s.

n +

Therefore a.s.

X~,,,--+X~

on

E ~ (m,n--+oo).

Since

oo

_<

P ( E k ) _< k=l

k=l

by the arbitrariness of e, we obtain lim

rn,nEN+

X,~,, = X ~

a.s.

C o r o l l a r y 3.1 Let B be a Banach space. Then the following statements are equivalant: (i) B has Radon-Nikodym property, (ii) For any two-parameter B valued martingale { X ~, j r , z E N~ } satisfying sup~eN ~ E IIX ~]I log + ]lXziI < ~ , there exists an integrable B valued random variable X ~ such that lim~eN~ X~ Xc~

a.8.~

(iii) For any uniformly bounded two-parameter B valued martingale {X,, ~'~,z E N_~}, there exists an integrable B valued random variable X ~ such that lim=eN~ X~ = X ~ a.s., (iv) For any uniformly bounded two-parameter B valued martingale { X~, :Tz~,z E N~ }, there L1

exists an integrable B valued random variable X ~ such that Xz -+ X ~ .

A. S. Convergence of Two-Parameter Banach Space Valued Martingales and RNP of Banach Spaces

195

Proof (i)=>(ii) follows from T h e o r e m 3.1. (ii)~(iii) and (iii)~(vi) are obvious. (vi)=>(i) follows from T h e o r e m 6 of [3, p. 127]. T h e o r e m 3.2 Let { Z , , J r , , z e N ~ } be a B valued martingale and let { ~ , z the local F4 condition. (i) For any A > O, then he(x*

e

> ~) < - -

-e--1

e N~} satisfy

sup SllX~lllog + IIX~ll + 1),

zEN~.

(ii) For any p > 1, then

E(X*) p <

sup E[IX~I[ p. z6N~_

Here X* = lim, eN~ IlX~ll. Proof (i) By L e m m a 1.3 we can choose sequences a,~ j" a and zn t (e~, oo) such that G,~ 6 9c~. and {5c~, z _ zn} satisfies the F4 condition on G,~ for all n. It follows from [2, T h e o r e m 2.8] t h a t {gv~ [a~ z _ z.} satisfies the F4 condition. Since for any z' > z > zn, we have E(X~,Ia. 1~

[a.) = E(X~, I .T~)Ia. = X ~ I a . ,

and X ~ I a i s 2~ Iv. measurable, { X ~ I G ~ , ~ ]a~, z >_ z,~} is a B valued martingale. It follows from L e m m a 3.1 t h a t for any A > 0, then AP ( s u p []X~[[IG~> A ) < e_ (sup \z>_z. - e 1 \z_>~

EI[X~llIG~log + IlX~ll/c. + 1/ 9

Hence

AP({X* > A} N

) _<

sup EllXzlltog + [[X~[I + 1)

<- e e- l ( SeNPEllX lll~

llX ll + l ) .

Letting n --~ oo we get ~ P ( X * > ~) <

e ( s u p E[]X~[[log+[IX~I[+I I . e - 1 \z6N~

-

-

In a similar way we can prove (ii). T h e o r e m 3.3 Let B be a Banach space with the Radon-Nikodym property and let {X,,.T',, z e N~} be any B valued martingale and {bc,, z e N~} satisfy the local F4 condition and

sup El[X~l[log + IlX~l] <

~.

~eN~

Then there exists an integrable B valued random variable Xoo such that lim X~ = Xoo ~eN~

a.s.

196

Shizin Gan

Proof Since {gcz, z E N~_} satisfies the local F4 condition, there exists a sequence {G,~} with G~ N Gj = 0(i y~ j), D~ G. = ~ and for each n, 3z. E N~ such that G . E 9vz. and {gvz, z > zn} satisfies the F4 condition on G~. It follows from [2, Theorem 2.8] that {~'~ ]Gn, z > z~} satisfies the F4 condition. Obviously {XzIG,~, . ~ [a,~,z > zn} is a two-parameter B valued martingale satisfying sup EliX~IG ~ It t~ + IIX~/c. II < oo. z6N~

It follows from Theorem 3.1 that {X~Ian,z >_ z,~} a.s. converges to integrable B valued random variable X a ~ I a . Therefore {X~, z E N~_} a.s. converges to ~-']~176Xa~Ian d J Xoo. Integrability of Xoo follows easily from the Fatou's lemma. C o r o l l a r y 3.2 Let B be a Banach space. Then the following statements are equivalent: (i) B has Radon-Nikodym property, (ii) For any two-parameter B valued martingale {X,, 3=~,z E N~} where { f z, z E N~} satisfies the local F4 condition and {X~, z E N~} is L log + L bounded. Then there exists an integrable B valued random variable Xoo such that lim~eN~" Xz = Xoo a.s., (iii) For any uniformly bounded two-parameter B valued martingale {X~, ~r, z E N~}, where { ~ , z E N~} satisfies the local F4 condition, then there exists an integrable B valued random variable X ~ such that lim,~N~ X~ = Xoo a.s., (iv) For any uniformly bounded two-parameter B valued martingale { X ~ , ~ z , z E N~}, where { 3c~, z E N~ } satisfies the local F4 condition, then there exists an integrable B valued L1

random variable Xoo such that Xz --+ X ~ .

Remark All theorems and corollaries in this paper can be easily extended to the case of any n-parameter B valued martingales (n _> 2).

References [1] [2] [3]

R Caioli, T B Walsh. Stochastic integrals in the plane. Acta Math, 1975, 134:111-183 Xingwu Zhuang, Jitao Li. The local F4 condition and the a.s. convergence of two-parameter martingales. Acta Math Sinica, 1987, 30(3): 412-418 J Diested, Jr J J Uhl. Vector measures. Amer M a t h Soc Surveys, Rhode Island providence, 1977, 15