Proc. Natl. Acad. Sci. USA Vol. 73, No. 6, pp. 1798-1799, June 1976 Mathematics On vector-valued asymptotic martingales (convergence almost everywhere/infinite-dimensional Banach spaces) ALEXANDRA BELLOW Department of Mathematics, Northwestern University, Evanston, Illinois 60201 Communicated by Alberto Calderon, March 15, 1976 ABSTRACT In contrast to the vector-valued martingale sumed that (an )ne N is the "minimal" sequence, that is, rn = which converges strongly almost everywhere if and only if the .. for each n e N. Banach space E has the Radon-Nikodym property, the vector- a(X1,X2, ,X,) valued asymptotic martingale converges strongly almost ev- The key tool in the proof is the following beautiful lemma erywhere if and only if E is of finite dimension. on which Dvoretzky and Rogers based the proof of their cele- brated theorem on absolute and unconditional convergence in 1. Let (9,7,P) be a probability space. Let N = 11,2,3,.. .4 and Banach space (see ref. 4); the following weak form of the let (On)neN be an increasing sequence of sub-a-algebras of Ar, Dvoretzky-Rogers lemma suffices for our purposes: i.e., if n < m, then 5'n C S. A bounded stopping time [with LEMMA (Dvoretzky-Rogers). Let E be a Banach space respect to the sequence (an)ne NI is a mapping r: Q - N such of infinite dimension. Let XI > 0...., Xn > 0 be any that jr = nI e 5n for all n e N, and r assumes only finitely given positive numbers. Then there exist distinct vectors many values. Let T be the set of all bounded stopping times. e, ...,en eE with lleill= lfori= 1,...n and such that if With the definition r < a if T(w) < a(Z) for all w e Q, T is a A' denotes the sum over any subset of I1, . nI, then directed set "filtering to the right" (note that if r e T, a e T then r V a e T). I-'AejiI2 < 32'A 2. [1] Let E be a Banach space. Let Xn: - E for each n e N. We may now state and prove our theorem. The sequence (Xn)ne N is called adapted if Xn: Q E is THEOREM. For a Banach space E the following assertions Bochner 97n-measurable for each n e N. are equivalent: The notion of asymptotic martingale emerges as an im- (1) E is offinite dimension. portant notion in the last few years (see refs 1-3, 5, 6). We recall (2) Every E-valued asymptotic martingale (Xn)n, N such its definition: that Supr T £ IIXT || < x, converges to a limit strongly almost Definition: An adapted sequence (Xn)ne N of E-valued everywhere. random variables is called an E-valued asymptotic martingale (3) Every E-valued asymptotic martingale (Xn)nE N such ifXn is Bochner integrable, i.e. IIXn(W)IIdP(o) < o for each that IIXn(w)I < 1 for each n e N and w e ( converges to a n e Nand limit strongly almost everywhere. Proof: Let (SXT)CET converges in the norm topology of E. (1) ) (2). u1, .... I be a basis for E. Then we can write for each n E N and (A e ( We recall the fundamental almost everywhere convergence X,(W) = X,'(w)ul + - + X,,'(w)u, theorems for asymptotic martingales: THEOREM I (Real-valued case). Let (Xn)n N be a real- and it is easy to check that for each 1 S j < r, (Xjn)nE N is a valued asymptotic martingale and suppose that SUpne N IXnI real-valued asymptotic martingale with SUpn, N fIXij < x. < co. Then (Xn)n & N converges to a limit almost everywhere Then clearly strong convergence almost everywhere of (Xn)ne N (see ref. 1). is equivalent to convergence coordinate-wise and this follows THEOREM II (Vector-valued case). Let E be a Banach space from the real-valued case (Theorem I). having the Radon-Nikodym property and a separable dual. (2) ) (3) is obvious. Let (Xn)n, N be an E-valued asymptotic martingale such that (3) ) (1). Suppose that E is of infinite dimension. We shall SupTe T SIIXI11 < O. Then (Xn)n, N converges to a limit weakly then construct an E-valued asymptotic martingale (Xn)n N almost everywhere (see ref. 3). with IIXn(co)II = 1 for each n e N and w E Q, which cannot Examples were constructed in ref. 3 for the case of the Ba- converge strongly to a limit almost everywhere. nach space E = 1p (1 < p < co) showing that weak convergence For each integer n > 1, let{A (n)j, 1 <i < 2n be the dyadic cannot be replaced by strong convergence. However, the case division of order n of [0,1), that is of a general Banach space E remained open. Recently this question was explicitly raised by Prof. Louis Sucheston in his < i < talk at the San Antonio Meeting (January 1976): For a Banach -([i 21 2 2n space E, is it true that every E-valued asymptotic martingale Apply now the Dvoretzky-Rogers Lemma to XA(n) = 1kn for that < c al- (Xn)ne N such sup,,Tf IIXTII converges strongly 1 < i < 2n. There exist then distinct unit vectors ej(n), 1 < i < most everywhere if and only if E is of finite dimension? in such that the is satisfied. Define now (Apparently this conjecture is due to Gilles Pisier.) The purpose 2n, E, inequality [1] of this note is to give an affirmative answer to the above ques- tion. X,1(U) = Z1.4,nuc)eil, for ccE(. 2. For simplicity we assume in this section that Q = [0,1), 9r 2=1 = the a-algebra of Borel sets, and P = Lebesgue measure. When Then clearly gn = a-(X1, nX) is the algebra spanned by the the sequence of a-algebras is not explicitly mentioned, it is as- atoms jAt W), 1 < i < 2 n. 1798 Downloaded by guest on October 2, 2021 Mathematics: Bellow Proc. Natl. Acad. Sci. USA 73(1976) 1799 We next show that Suppose now that the asymptotic martingale (Xn)ne N did converge to a limit strongly almost everywhere: 'a Ae U =4 |X T O . [2] nElN A JT lim Xn(w) = X_(w) strongly for almost everywhere wEQ. nGIN This of course implies that (X,,), N is anaymptotic martin- (2) gale. For this purpose, let A e 9No. Take Ee T, X > No; there By the Lebesgue Dominated Convergence, (2) implies is then an integer K such that NO-< T < K. For each No < n < K, JXn f x-9 foreachACG7. (3) IT = ni n A = U'Aj fn = Qn From (1) and (3) we deduce and thus fX- = 0 for all AGC, c K = K and since XC. is 9-measurable, this implies JX fT fA ( j A nejIn) = nNO£eP(Aj(n))ej(n) X,(W) = O for ahnost everywhere wEQ. By inequality [1], we have for each No < n < K But this contradicts (2), since JIX,( w)II1 = 1 for each n e N and 1 (A(n je(n)||< .(PA(n))2)/ w e Q. This completes the proof of the theorem. Note Added in Proof. A similar argument shows that for a Banach = = (/)f(P(Qn ))1/2 < space E, the class of E-valued asymptotic martingales (X,), N such (zE'P(A,'))) that SUpneN fIIXn < X and the class of E-valued asymptotic mar- tingales (Xn)n, N such that SUPre Tf IIX7 11 < o coincide if and only We deduce if E is finite-dimensional. < 1. Austin, D. G., Edgar, G. A. & Ionescu Tulcea, A. (1974) "Pointwise XI| E || P(A.()) || < r3 e~)<e convergence in terms of expectations," Zeit. Wahrscheinlich- keitstheorie verw. Gebiete 30, 17-26. provided No is large enough (so that 2n2No VA/(v2)n <E). 2. Bellow, A. (1976) "Stability properties of the class of asymptotic Thus, [2] is proved. In particular then, for each A & Uni N Ym, martingales," Bull. A.M.S., 82, 338-340. 3. Chacon, R. V. & Sucheston, L. (1975) "On convergence of vec- we have tor-valued asymptotic martingales," Zeit. Wahrscheinlich- keitstheorie verw. Gebiete 33,55-59. 4. Dvoretzky, A. & Rogers, C. A. (1950) "Absolute and unconditional IA Xn O convergence in normed linear spaces," Proc. Natl. Acad. Sci. USA 38, 192-197. Now since 1IXn(co)II = 1 for each n e N and w e Q, it follows 5. Dvoretzky, A. (1976) "On stopping time directed convergence," that Bull. A.M.S. 82,347-349. 6. Edgar, G. A. & Sucheston, L. (1976) "A marts: A class of asymptotic AG Y = or U aid) J~Xn °.(1) martingales (Discrete parameter)," J. Multivariate Analysis, in "EIN press. Downloaded by guest on October 2, 2021.