Cramér large deviation expansions for martingales under Bernstein's condition

Xiequan Fan (1) , Ion Grama (1) , Quansheng Liu (1)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales.

Mots clés: expansions of large deviations; Cramér type large deviations; central limit theorem; large deviations; moderate deviations; exponential inequality; Bernstein's condition

Domaines: Probabilités [math.PR]

Xiequan Fan, Ion Grama, Quansheng Liu. Cramér large deviation expansions for martingales under Bernstein's condition. Stochastic Processes and their Applications, 2013, 123, pp.3919-3942. ⟨hal-00905517⟩