Large deviation inequalities for martingales in Banach spaces

TL;DR

This paper establishes large deviation inequalities for martingales in Banach spaces, providing bounds on the probability that the partial sums exceed a threshold, under conditions of exponential or polynomial tail decay.

Contribution

It introduces new upper bounds for martingale partial sums in Banach spaces under exponential and polynomial tail decay conditions.

Findings

Bounds are derived for exponential tail decay case.

Bounds are derived for polynomial tail decay case.

Results extend large deviation inequalities to Banach space martingales.

Abstract

Let $(X_i, \mathcal{F}_i)_{i\geq1}$ be a martingale difference sequence in a smooth Banach space. Let $S_n=\sum_{i=1}^nX_i, n\geq 1,$ be the partial sums of $(X_i, \mathcal{F}_i)_{i\geq 1}$. We give upper bounds on the quantity $\mathbb{P}\left(\max_{1\leq k\leq n}\lVert S_k\rVert>nx\right)$ in terms of $ n\geq 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.