Large deviation inequalities for martingales in Banach spaces

Abstract

Let $(X_i,{\mathcal{F}}_i{)}_{i\ge 1}$ be a martingale difference sequence in a smooth Banach space. Let $S_n={\sum }_{i=1}^n{X}_i,n\ge 1,$ be the partial sums of $(X_i,{\mathcal{F}}_i{)}_{i\ge 1}$. We give upper bounds on the quantity $\mathbb{P}\left({\max }_{1\le k\le n}\Vert S_k\Vert >nx\right)$ in terms of $n\ge 1$ and $x>0$ in two different situations: when the martingale differences have uniformly bounded exponential moments and when the decay of the tail of the increments is polynomial.