Large Deviations of Vector-valued Martingales in 2-Smooth Normed Spaces

TL;DR

This paper establishes exponential bounds for large deviations of vector-valued martingales in finite-dimensional normed spaces, emphasizing dimension-independent bounds when the space's norm approximates a differentiable norm with a Lipschitz continuous gradient.

Contribution

It introduces dimension-independent exponential bounds for martingales in certain normed spaces, expanding understanding of large deviations in these contexts.

Findings

Dimension-independent bounds are achievable in spaces with differentiable norms.

Examples of spaces with the required norm properties are provided.

The bounds are nearly dimension-free under specific norm approximations.

Abstract

We derive exponential bounds on probabilities of large deviations for "light tail" martingales taking values in finite-dimensional normed spaces. Our primary emphasis is on the case where the bounds are dimension-independent or nearly so. We demonstrate that this is the case when the norm on the space can be approximated, within an absolute constant factor, by a norm which is differentiable on the unit sphere with a Lipschitz continuous gradient. We also present various examples of spaces possessing the latter property.