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

Anatoli Juditsky (LJK); Arkadii S. Nemirovski (ISyE)

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

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arxiv 0809.0813 v2 pith:DNHH5MPD submitted 2008-09-04 math.PR math.STstat.TH

Anatoli Juditsky (LJK) , Arkadii S. Nemirovski (ISyE) This is my paper · ORCID

classification math.PRmath.STstat.TH

keywords spacesboundscasedeviationslargemartingalesnormnormed

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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.

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