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

Anatoli Juditsky, Arkadii S. Nemirovski

Published 2008-09-04, updated 2023-01-30Version 2

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.