Since (s(x*, Fn(co)nG(co)), n > 1) is equicontinuous for the Mackey topology, ...... A. Lechicki and S. Levi, Wijsman convergence in the hyperspace of a metric ...
transactions of the american mathematical society Volume 341, Number 2, February 1994
ON CONVERGENCE AND CLOSEDNESS OF MULTIVALUEDMARTINGALES ZHEN-PENGWANG AND XING-HONG XUE Abstract.
In this paper, various convergence theorems and criteria of closed-
ness of multivalued martingales, submartingales, and supermartingales are proved.
1. Introduction
and preliminaries
The study of multivalued functions has been developed extensively with applications in several areas of applied mathematics, such as mathematical economics, optimal control, and decision theory, (cf. Hildenbrand (1974), Him-
melberg and Vleck (1974), Papageorgiou (1986), Clarke (1984), de Korvin and Kleyle (1985), Vovits, Foulk and Rose (1981), and Aubin and Frankowska (1990)). Four notions of convergence of multivalued functions in a Banach space, the Hausdorff distance convergence, the Kuratowski-Mosco convergence, the weak convergence, and the Wijsman convergence are particularly useful in the study. Illustrated by the works of de Korvin and Kleyle (1985) and Papageorgiou (1986), multivalued martingales, submartingales, and supermartingales are powerful tools in the study of convergence of random multivalued functions. In this paper we shall make a further study on convergence and closedness of multivalued martingales, submartingales, and supermartingales. Throughout this paper (fi, sé , P) is a complete probability space, X is a separable Banach space with the dual X*, and 2X is the set of all subsets of
X. Let PC(X) = {A £ 2X: A is nonempty, closed, and convex}, Pcb(X) = {A£ PC(X): A is bounded},
PwkAx) = {Ae PC(X): A is weakly compact}. For A e 2x\(f), we denote by clA and coA the closure and the closed convex hull of A respectively, and define \A\ = sup{||.x|| : x e A} ,
s(x*, A) = sup{(x*, y) : y e A},
d(x, A) = inf{\\x -y\\: y e A},
s(x*, 0) = -oo,
d(x, i>)= oo,
x* e X*,
xeX.
Received by the editors December 19, 1991.
1991MathematicsSubjectClassification.Primary 60G42,60G48,60D05. Key words and phrases. Multivalued martingale, submartingale, supermartingale, KuratowskiMosco convergence, Hausdorff distance convergence, weak convergence, Wijsman convergence, closedness of martingales, uniformly integrable, uniform sequence of subpramarts. Research of the first author supported by the National Science Foundation of China. © 1994 American Mathematical Society
0002-9947/94 $1.00+ $.25 per page
807
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808
ZHEN-PENG WANG AND XING-HONG XUE
s(x*, A) and d(x, A) are called the support function and the distance function of A respectively. Let !F be a sub-a-algebra of sé . A random multivalued function F from fi -> PC(X) is ^"-measurable, if there exist y-measurable random variables fn : fi -> X,
F(œ) = cl{fn(œ),n>l},
wGfi,
(cf. [4, Chapter III]). The original study of random multivalued functions goes back to Robbins (1944, 1945). Let Jt[9~] be the family of all random multivalued functions which are ^-measurable. Let yc = {F e Jf[sé\: ywkc = {Fey:
ydx = {F £yc:Ed(0,F)) £ F (to) a.s.},
where Lxx is the set of all Bochner integrable r.v.'s from fi —► X. We also simplify SxF(sé) as SF . It is easy to show that SF is nonempty if and only
if F £ ycdx. The Aumann integral EF of F is defined by EF = {Ef: f £ SF}, where Ef is the Bochner integral of /. The conditional expectation of F £ ydx with respect to &, E(F\9'), is the unique (up to a P-null set) ^"-measurable multivalued function in ydx such that
sxE(Fm(y) = d{E(f\yy.f£SxF}, the closure in L\, (cf. [15, Theorem 5.1]). The conditional expectation of a random multivalued function behaves like the traditional single-valued conditional expectation. For example, for any F e ydx, E(F\^) = F a.s., if F is •^"-measurable; and for any sub-er-algebras yx,^2 of sé ,
E(E(F\9x)\92) = E(F\92)a.s.,
if^c^,
(cf. [15, Theorem 5.3]). For details of the definitions and properties of the measurability, the integration, and the conditional expectation of random multivalued functions, we refer the reader to the works of Castaing and Valadier ( 1977),
Hiai and Umegaki (1977), Hiai (1985), Papageorgiou (1985a), and Aubin and Frankowska (1990). Yor A,B e2x, let h+(A,B) = sup{d(a, B): a e A}, h~(A, B) = sup{d(b,A): b e B}. The Hausdorff metric h on PC(X) is de-
fined by h(A, B) = max{A7+(^, B), h~(A, B)},
A, Be PC(X).
For F, G Gycx, define A(F, G) = Eh(F, G). Then (PC(X), h) and (yx, A) are complete metric spaces (cf. Hiai and Umegaki (1977), p. 160). For (An, n > l,A)c PC(X), let 5-liminf/i„ = {x £ X: limúí(x, A„) - 0} , and it;-lim sup ^„ = {x £ X: X/c^ x, for some xk £ A„k},
where (aî^ , k > 1) is a subsequence of (aa > 1) and A means convergence in the weak topology of X. We denote by A„ -^ A the convergence of An
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MULTIVALUED martingales
809
to A in the Kuratowski-Mosco sense, if w;-limsup.4„ = A = s-hminfv4„ ; by An —► A the Hausdorff distance convergence, if h(A„ , A) -* 0 ; and by A„ -> A the weak convergence, if s(x*, A„) -» s(x*, A) for each x* e X*. We say that An is Wijsman convergent to A if d(x, A„) -* d(x, A), xel,(cf. [10 and
19]). Since h(A, B) - sup{\d(x, A)-d(x, B)\, x e X} , the Hausdorff distance convergence is stronger than the Wijsman convergence. However, if (and only if) X is totally bounded, these two notions of convergence coincide (cf. [8 and 33]). If restricted to Pcb(X), then the Hausdorff distance convergence implies the Kuratowski-Mosco convergence and the weak convergence, and when X is a finite dimensional space, these three types of convergence are equivalent (cf.
[13, p. 165] and [30]). Clearly, for any Heyx,
H(œ) e Pcb(X) a.s.
Let (yn, n > 1) be an increasing sequence of complete sub-cr-algebras of s* , ^o
= o{\Jn&ñ) ■ By (Fn,yn,n>l)
we mean that (Fn) c ydx
and F„
is ^¡-measurable, aa > 1. Let T be the set of bounded stopping times with respect to L% , n > 1) and T(s) = {teT:t>s}, seT. Definition 1.1. (F„ , SFn, n > 1) is called a (multivalued) martingale, submartingale, or supermartingale, if
E(Fn+x\yn) =,D,
or cF„ a.s.,
n > 1.
The rest of this paper is organized as follows. In §1, we shall prove various convergence results and criteria of closedness of multivalued submartingales. In §§3 and 4, we shall focus on convergence and closedness of multivalued martingales and supermartingales.
2. On multivalued
submartingales
Lemma 2.1. For any F e ydx and sub-a-algebra SF c sé ,
(i) \E(F\S?)\ < E(\F\ \y) a.s.; (ii) [12] d(x,E(F\y)) 1} a.s. (cf. [15, Lemma 1.1]). It follows that for any x e X, x* e X* and e > 0, there exists f e SF such that d(x, F) > \\x - /|| - e a.s. and s(x*, F) < (x*, f) + s a.s., and we can easily
check that \E(F\&-)\=
esssup
|/| = esssup|£(/|y)|
essinf ||jc - £(/|^)|| ' fts'r
= d(x, E(F\y))
a.s.,
= essinf£(||;c - /|| \9~) =
essinf f^Em^)
a.s.,
xeX,
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\\x - f\\
ZHEN-PENG WANG AND XING-HONG XUE
810
and for each x* e X* s(x*, E(F\3r)) =
ess sup (x*, f) = ess sup(x*, E(f\F))
= esssupE((x*,f)\y)
= E\esssup(x*,f)\Sr)
/€5} V f^SiF = E(s(x*, F)\9~) a.s. Q.E.D.
J
The following lemma slightly modifies Egghe's lemma [9, Lemma VIII. 1.15]. Lemma 2.2. Let I be a countable set and I(n) a subset of I such that I(n) c 7(aa + 1) and \Jn>xI(n) = I. (a) {(x'n, 5*~n,n > m), uniform sequence of real-valued subpramarts, i.e.,
lim sup P >£TseT(t)
sup [x\ - E(x's\&¡)] > e \i€i(t)
i e I(m), m > 1} is a
= 0,
e > 0,
I
if and onlyif (2.1)
limP [ sup (x¡ - r¡) > e =0,
teT \iei(t)
J
e>0,
where I(t) = I(n) on (t = n), and ri. —essinf E(x't\âr„) a.s., ter(n)
i G I(n) ,n>l.
(b) Let {(x'„,S*¡,, n > m), i e I(m), m > 1} be a uniform sequence of real-valued subpramarts such that liminf£ "
sup \x'„\ < oo. i€l(n)
Then for each m > 1 and i e I(m), (x'„, n > m) and (r'n, n > m) converge a.s. to integrable r.v. 's xl and r' respectively, x' = r' a.s., and
(2.2)
lim I sup x'„ = sup lim x'n = sup*'. " \iel{n) J ('€/ " i€l
Proof. For each m > 1 and i e I(m), it is well known that (r'n, 5*¡¡, n > m) is a generalized submartingale with values in [-00,00), and it is easy to show that for any fixed t e T and i e I(t), r\ = essinfser(i) E(xlsffi) a.s. and there exist (í¿, n > 1) c T(t) such that E(x'i\9t) | r\ a.s., (cf. [6, Lemma 4.1 and
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MULTIVALUEDMARTINGALES
811
Theorem 4.1]). Hence, for any e > 0,
lim sup P f sup[x\ -E(xí\9¡)]> e\ t£T s€T(t) \i€l(t)
J
< limP [ ess sup I sup [x\ - E(x\\3^)] ) > e ]
^T \
sent) \iei(t)
J
J
= limP ( sup ( ess sup [x\ - E(x's\S*i)]) > e ] = limP
t*T V/6/(i)V sent)
J
J
ter
sup [x\ - r't] > e I
\iei(t)
I
= limP [ sup ( sup[jc/ - E(x', \9t)]\ > e 'er \/6/(i) J , ier \/e/(i) V n = lim P ( sup ( sup [x\ - E(x{, \&t)]\ > e ]
^T
\n
\iel(t)
"
J
J
= lim sup P ( sup [jc/ - E(xt, \9¡)] > e ] t£T
n
\iel{t)
"
/
< lim sup P f sup [jc/- E(x¡\SC)]> e ) , '£Tsent)
\iei{t)
J
where the last equality is based on the fact that (supi6/(n[x/ - E(x\ \&¿)], n > 1) is increasing, and (a) holds. Now assume that {(xln,y„, n > m), i e I(m), m > 1} is a uniform sequence of real-valued subpramarts such that lim inf„ E sup,€/(„) |jc¿| < oo . Since (2.1) is equivalent to (2.3)
sup \x'n-r'n\->0 a.s. '€/(«) (cf. [20, Theorem 5.1]), applying Millet and Sucheston's subpramart convergence theorem [20, Theorem 5.1] and (2.3), for each i e I(m), m > 1, (x'„, n > m) and (r'„, n > m) converge a.s. to integrable r.v.'s x' and r' respectively, xl = r' a.s. To prove (2.2), first we assume that inf„>i,6/(„)Xi > a for some a < 0.
Then inf„>1(€/(„)r¿ > a and liminf„ ¿s sup,6/(„) \r'n\ < liminf„ ¿? sup,e/(n) |jc¿| + \a\ < oo. Applying the proof of Neveu's (1972) lemma [21, Lemma 4, or 22, p. 109], lim„ supie/(n) r'n = sup;6/ lim„ rn = sup;€/ r', and
by (2.3), lim sup x'„ = lim sup r'n - supr' = supx', " i€l(n)
" iei(n)
iel
;'£/
(2.2) holds. In a general case, for any fixed a < 0, since
P f sup [jc/Va - E(x's Va\9Ç)}> e ] < P [ sup [x\V a - E(x¡\&¡) Va]>£
\iei(t)
J
\iei(t)
e)
\tei(t)
J
J
,
{(x'„ V 0,3%, n>m), i e I(m), m > 1} is a uniform sequence of real-valued subpramarts satisfying liminfis "
sup |jc¿ V a\ < oo. iel(n)
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812
ZHEN-PENG WANG AND XING-HONG XUE
Thus sup jc' < lim sup x'n < lim sup (x'n V a) iel
" iel(n)
n iel(n)
—suplim(jc¿ V a) = sup(jc' Va)-> sup jc' , /€/
"
¡el
iei
as a -» -oo, (2.2) holds. Q.E.D. Definition 2.1. Given (a) or (b) holds: (a) ywkc > liminf« h+(F„ ywkc, h(F„ , GnFn)
(F„) c -S£ , we say that (i) assumption (A) holds, if either X has the Radon-Nikodym property, (b) for some G e , G) = 0 a.s.; (ii) assumption (B) holds, if for some G e —► 0 a.s.; (iii) (Fn) is L'-bounded, if supnE\Fn\ < oo .
Let JV be the set of all positive integers, JV"2= {(m, k): m eJV, k eJV}, and J^2(l) = {(m,k)eJr2: m < 1}, /> 1. Lemma 2.3. Let (Fn,5*„,n> 1) be a submartingale. Then there exists a family of adapted sequences {(fim' ', 3%, n > m), (m, k) e J^2} c Lxx such that
(a) for each n > 1, Fn(co)= d{jf'k)(io),
k > 1} = cl{jim>k)(co), lm,k>l,
and {(||/im'fc)|| ,Srn,n>l),(m,k)e yf2(l), I > 1} is a uniform sequence of subpramarts. (b) under assumption (A), if sup„ ¿^„l < oo, then there exist r.v. 's (fîm-/(«.*) limsupd(x,
a.s.,
Fn) < d(x, F) a.s.,
xeX,
n
and (2.9) and (2.10) in Theorem 2.1 below hold, where (2.6)
F = cô{f{m'k),(m,k)eyr2}eycx.
Proof. Choose {fî"'k), k > 1} c SxFn(yn)such that
Fn(co)= cl{fk)(a>),k>l}. By the definition of multivalued submartingales and conditional expectations, S\n(yn) C SxE(Fn^n)(yn) = cl{E(f\3rn):feSxFnjyn+x)}.
Since An'k) G SFn(9n), there is jf+\k) e SxFnjyn+x),
E\\Ún'k)-E(&+\k)\^n)\\< >n+l I »)»
— 2n+k+n+l'
For each k > 1 and m > 1, by induction, we can get fim'k) £ SFn(y„), n > m , such that
/^.*>-*(j£.*>|SÇ)||< 2m+k+n+\ * License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
'
MULTIVALUEDMARTINGALES
Hence, for each (m, k) e Jf2,
(Am'k) ,^n,n>m)
813
is a quasi-martingale i.e.,
En>mE\\fÍm-k) - EU%ik)\9-R)\\ < oo . For any j > 1, t e T(j) and s e T(t), let K —maxweciS(co), then (s = K) e S\-\,
E\\Am'k)-E(fk\yK_x)\\I(t = n)
n
and
(\\^k)\\-E(\\fjm-k)m))>e)
sup ^m ,k)eyr2(t)
k)\yn)\\I(t = n)>e) m7 fc>l mk)\?T)\\= 0 i^TseT{t)
(Bellow (1978)). If X has the Radon-Nikodym property, (2.4) follows from Bellow's (1978) uniform amart convergence theorem. Now we assume that for some G e ^fwkcliminfn h+(F„, G) = 0 a.s. By the Riesz decomposition theorem for uniform amarts (cf. [3 and 11]), JÍm'k) = h„m'k) + znm'k), where (hnm' \3%,n > m) is a martingale and lim„zim' * = 0 a.s. and in Lx.
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814
ZHEN-PENG WANG AND XING-HONG XUE
Hence (h„m'k) ,5*¡,,n>m) is an Lx-bounded martingale and for a.s. œ e fi, (h„m' \co)) has a weakly accumulative point in G(co). Applying [5, Proposition 4.4], h„m'k), and hence f¡m'k), strongly converges to /) = d(x, F)
a.s.,
(2.5) holds. By Lemma 2.2, lim|F„|=lim n
sup "
=
\\fi„m'k)\\
(m,k)e^2(n)
sup
lim\\JÍm'k)\\=
{m^ejr"2
sup
n
\\f[m'k)\\
[rn,k)e^2
sup
||a/ 1} is a uniform
\x„m'k)\ s lx*, f\E(Fn\^))
n
>s(x*,E(F\$-))
a.s.,
(4.5) holds. If X* is strongly separable, then (4.6) follows from (4.5). If fi« E(Fn\9-) e 5?wkc a.s., then f)n E(FH\f) D E(F\F) e 5fwkc a.s., and by the continuity in the Mackey topology, (4.5) implies (4.6). Q.E.D.
Definition 4.2. Given (F„) c 2Cdx, we say that (i) assumption (Aj) holds, if X* is strongly separable and if there is a subsequence {nk} such that F„k e S'wkc, k > I ; (ii) assumption (B'j) holds, if Fn e 2Cwkc, n > 1. Given a supermartingale
(Fn , S?n, n > 1), we define G„ — f)m>n E(Fm\&„).
Lemma 4.3. Let (Fn,&~„, n > 1) be a supermartingale satisfying assumption (A'[) or (B',). Then (Gn , SFn, n > 1) is a martingale. Proof. By the definition of supermartingales, we have FnDE(Fn+x\^n)DE(Fn+2\^n)D---
a.s.,
n > 1.
If assumption (B',) holds, then, by Lemma 4.2, G„ e £?wkc and E(Gn+i\&n) = E ( fi E(Fm\^H+i)\9n\ \m>n+l J
= (] E(Fm\^„) = G„ a.s. m>n
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MULTIVALUEDMARTINGALES
823
In the following we assume that X* is strongly separable and there exists a subsequence (nk) such that F„k e £?wkc, k > I. Then, for any Ac> 1, f]m>nk E(Fm\&nk) e S?wkc■ For any n > 1 , choose nk-X > n , then, by Lemma
4.2," *
G„=f] E(Fm\9n)= fi E(E(Fm\9-nk_xWn) m>n
= e(
tn>"k-¡
fi
E(Fm\rnk_t)\9-n)esecdx,
\m>nk-x
j
and
E(Gn+i\9n)= E ( fl £(FW|^+1)|^ \m>n+l
y
= E j f| ¿í(¿í(Fm|^J|^¡+1)|^¡
=WWn ^(^mi^)i^¡+i 1^ = ¿7( f| F(FW|^J|^¡ J =: fl E(E(Fm\9nk)\9n) \m>ni¡
J
m>nk
= f) ¿?(Fm|^) = G„ a.s. (C7„, ^ , aa> 1) is a martingale. Q.E.D. For a supermartingale (i7,,, &~„,n > 1), by Lemma 2.1, (d(x, F„), &~„, n > 1) is a real-valued submartingale. Let v„ = supOT>„¿?(o'(0,Fm)\&„). It is well known that (vn, n > 1) is the martingale part in the Krickeberg decomposition of submartingale (d(0, F„), S?„, n > I), lim„Evn = limnEd(0, Fn), and (vn, n > 1) is uniformly integrable if and only if (d(0, Fn), n > 1) is uniformly integrable (cf. [6 and 22]). In the following we assume that (F„,^,, n> I) is a supermartingale satisfying sup„ Ed(0, Fn) < oo, and use the following Hess' (1991) truncation: Fk = Fnr\B(0,vn+k),
k>l,
(since vn > d(0, Fn), Fk is not empty).
Lemma 4.4 (Hess (1991)). (Fk ,&n,n>l) is a supermartingale such that (i) sup„ ¿iii7*| < sup„ Evn + k < oo ; (ii) sup„ \Fk\ < sup„ v„ + k < oo a.s.; (iii) if
Fk K4*Fk, k > 1, then Fn K4*\jk Fk a.s.
Lemma 4.5. (i) If d(x, Fk) -> d(x, Fk) a.s., k>l, d(x,Fn)^d(x,\jFk\
then a.s.
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824
ZHEN-PENG WANG AND XING-HONG XUE
(ii) If liminf„ \F„\ < oo a.s., and if s(x*, Fk) -* s(x*, Fk) a.s., k > 1, then s(x*, F„) — s(x*, [\k Fk) a.s.
(iii) i/limsup„ \F„\ < oo a.s., and if Fk —► Fk a.s., k > I, then Fn —>(Jk Fk a.s. Proof, (i) It is easy to show that
limd(x, F„) < inflimrf(x, Fk) = infrf(jc, Fk) = d \ x, I \Fk ) k n
k
\
k
J
a.s.
On the other hand, we can choose x„ G F„ (xn is a function of co) such that d(x, F„) > ||x - x„|| - 1/aj . Then (||x - x„||, aa> 1) is bounded and
limd(x, Fn) = lim||x-x„||
> inflimiz(x, Fk) = d\x,\
■k n
\Fk\
V V
/
a.s.
(ii) Since Es(x*,Fn)- < Es(x*, Fk)~ < ||x*||ii(i;n + k), by Lemma 2.1, (s(x*, Fn), n > 1) and (s(x*, Fk), n > 1) are supermartingales and converge almost surely. If liminf,, |.F„| < oo a.s., then for a.s. co e fi, there exists k = k(co) > 1 such that for infinitely many aa> 1 F„(co) = Fk(co), and lims(x*, F„) = suplims(x*, n
k
Fk) = sups(x*, Fk)
n
k
= s(x*,U^ V
k
a.s.,
x* e X*,
j
(ii) holds, (iii) If limsup„ |.F,,| < oo a.s. and if Fk -* Fk , then the null set in the proof of (ii) can be independent of x*, and (iii) holds. Q.E.D. Proof of Theorem 4.1. By Lemmas 4.4 and 4.5, we may assume that sup„ E\Fn\ < oo, supn |F„| < oo a.s. and the following (1) or (2) is satisfied: (1) X* is strongly separable and assumption (A) holds; (2) assumption (B) holds. We may also assume that for some subsequence (nk), F„k e ^fwkc, k > 1, and in proof of (iii) liminf„ h+(Fn , G) = 0 a.s. for some G e ^wkc ■ Let G„ — f\m>nE(Fm\&n). Then (G„ ,&n,n> 1) is an L'-bounded martingale satisfy-
ing^ 1) or (2). Applying Theorem 2.1, there is F g 5fx such that (4.7)
G„*Fa.s.
and
Gn K^MF a.s.
For any n > 1, choose k > 1 such that nk > n. Then, by Lemmas 2.1, 4.1, 4.2 and the monotone convergence theorem for conditional expectations,
s(x*,Gn) = s[x\
f|
E(E(Fm\^nk)\9n)
= síx*,¿7[ fl E(Fm\?nkWn\ ) =£Îslx*, \
= E (lims(x',
\m>nk
j J
\
E(Fm\9-Hk)Wn) = HmE(s(x*,
\
fl E(Fm\9nk)\\9n\ rn>nk
E(Fm\9nk))\9n)
= lims(x*, E(E(FmWnkWn)) = H™s(x*, E(Fm\9n)) = limE(s(x* ,Fm)\9n)
a.s.,
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MULTIVALUEDMARTINGALES
825
and (s(x*, F„), SFn, n > 1) is a real-valued supermartingale satisfying supF|s(x*,
Fn)\ < ||x*||sup¿7|i7„| < oo.
n
n
Hence (s(x*, Fn), %, n > 1) is a amart, of course, subpramart (cf. Millet and Sucheston (1980)), and s(x* ,G„) = lim£(s(x*
,Fm)\9n) = ess inf E(s(x*, Ft)\^n)
m
teT(n)
a.s.
Applying (4.7) and Lemma 2.2, we have s(x*, F) = lims(x*, G„) = lims(x*, Fn) a.s., n
x* e X*,
n
(4.2) holds. As the proof of (b) in Theorem 2.1, we get (4.1) and (4.3) from (4.2), noticing that s- lim inf Fn D s-lim inf Gn = F n
n
a.s.
Proof of (iii). Since there is G G ^wkc such that liminf„h+(G„, G) < liminfnh+(Fn,
G) = 0 a.s., G D F e ¿^kc
a.s. (see the proof of Lemma
2.4). By Lemma 2.4 and its proof, limnd(x, Fn) < lim„iz(x, Gn) = d(x, F) a.s. and limd(x, n
Fn) > lim sup sup ((x*, x) - s(x*, Fn)) n
x'eB*
> sup ((x*, x) -s(x*,
F)) - d(x, F)
a.s.,
x'eM*
(iii) holds. Q.E.D. Proof of Theorem 4.2. Assume that for some subsequence (n^ , F„k e -2/We, Ac> 1. If (d(0, Fn)) is uniformly integrable, then (vn), and hence (\F^\), is uniformly integrable, /' > 1. Let G},k= f)m>nkE(F¿\^nk). Then, by Lemmas 4.4 and 4.3, (Gik, ^¡k, k > 1) is a uniformly integrable martingale satisfying assumption (A), and by Theorem 3.2, there is FJ e S?cx such that Gik = E(Fi\&nk) a.s. For each n > 1 choose k > 1 such that nk> n , then
FJDE(Fl\^H)DE(GÍk\^n) = E(E(Fi\FnkWn)
= E(Fi\Fn)
a.s.
Let F = (JjFJ. Then F e £?dx, and, by [14, Theorem 2.1],
(4.9) Fn= {JFj Del l IjE^ffî)]
=e(\JfJ\9„\
^ E(F\^n) a.s.
On the other hand, if (4.9) holds for some F e Jzfcdx,then, by Lemma 2.1,
d(0, Fn) < d(0, E(F\9n)) < E(d(0, F)\Pn), (d(0,Fn)) is uniformly integrable. Q.E.D. ACKNOWLEDGMENT
The authors are grateful to the referee for most helpful comments.
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826
ZHEN-PENG WANG AND XING-HONG XUE
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of Statistics,
East China Normal University,
Department
of Statistics,
Columbia University,
Shanghai 200062, China
New York, New York 10027
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