An Approach to Inequalities for the Distributions of Infinite-Dimensional Martingales

Abstract

Let (f _n)_n=0 ^∞ be a martingale (understood either in the weak or strong sense) in a separable Banach space (X, ∥ ∥) with respect to an increasing sequence of σ-algebras $(A_{n})_{n=0}^{\infty}, f_{0}\equiv 0, d_{n}=f_{n}-f_{n-1}, n=1,2,..., d_{0}\equiv 0, f^{*}=sup{||f_{n}||:n=0,1,...}, d^{*} denotes similarly, s=(\sum_{n=1}^{\infty}E_{n-1}||d_{n}||^{2})^{1/2}$, where E_n-1 stands for E(∣A _n-1).