On vector-valued amarts and dimension of banach spaces

Abstract

If (Xn)n~ N is an amart of class (B) taking values in a Banach space with the Radon-Nikodym property, then Xn converges weakly a.s., as proved in [4]. Examples exist in [4] and [7] which show that strong convergence may fail, but recently Alexandra Bellow [2] proved the following result: A Banach space E is finite-dimensional if (and only if) every E-valued amart of class (B) converges strongly a.s. We prove here that if p is fixed, 1 < p < o% then a Banach space E is finite-dimensional if (and only if) every LV-bounded E-valued amart converges weakly a.s. The point of this is that in the amart convergence theorem for an infinitedimensional Banach space, the assumption (B) cannot be weakened any more than the conclusion that weak a.s. convergence holds can be strengthened. Let (E2,~,P) be a probability space, N = { 1 , 2 . . . . }, and let (~),EN be an increasing sequence of a-algebras contained in ~ . A stopping time is a mapping r: s w { oo }, such that {7 = n} ~o~ for all n EN. The collection of bounded stopping times is denoted by T; under the natural ordering Tis a directed set. (The notation and the terminology of the present note are close to those of our longer article [7].) Let E be a Banach space and consider a sequence (X, ) ,~ of E-valued random variables adapted to (~),~N, i.e. such that X~: s is d -s t rongly measurable. We will write E X (expectation of X) for the Pettis integral [9] of the random variable X. The sequence (X,) is called an amart iff each X, is Pettis integrable and limr r EXr exists in the strong topology of E. An adapted sequence (Xn) is said to be of class (B) iff