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

Anatoli Juditsky(Anatoli.Iouditskiimag.fr)
Arkadi Nemirovski(nemirovsisye.gatech.edu)

Abstract: In this paper, 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 %, with a given constant, gradient. We also present various examples of spaces possessing the latter property.

Keywords: Large deviations, smooth normed spaces, vector-valued martingales

Category 1: Stochastic Programming

Category 2: Robust Optimization

Citation: Preprint, ISyE, Georgia Institute of Technology, 765 Ferst Dr. NW Atlanta GA 30332-0205

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Entry Submitted: 04/11/2008
Entry Accepted: 04/11/2008
Entry Last Modified: 04/11/2008

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