Conditional weak laws in Banach spaces

Abstract

Let (B, parallel to . parallel to) be a separable Banach space. Let Y, Y-1, Y-2,... be centered i.i.d. random vectors taking values on B with law mu, mu(.) = P(Y is an element of .), and let S-n = Sigma(i=1)(n) Y-i. Under suitable conditions it is shown for every open and convex set 0 is not an element of 2 D subset of B that P(parallel toSn/n - v(d)parallel to > epsilon\S/n is an element of D) converges to zero (exponentially), where v(d) is the dominating point of D. As applications we give a different conditional weak law of large numbers, and prove a limiting aposteriori structure to a specific Gibbs twisted measure (in the direction determined solely by the same dominating point).