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

Anatoli B. Juditsky (1) , Arkadii S. Nemirovski (2)

1 SMS - Statistique et Modélisation Stochatisque
2 ISyE - School of Industrial and Systems Engineering [Georgia Tech]

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.

Domaines: Probabilités [math.PR]; Théorie [stat.TH]

Anatoli B. Juditsky, Arkadii S. Nemirovski. Large Deviations of Vector-valued Martingales in 2-Smooth Normed Spaces. 2008. ⟨hal-00318071⟩