Self-normalized Cramér type moderate deviations for martingales

Xiequan Fan (1) , Ion Grama (2) , Quansheng Liu (2) , Qi-Man Shao (3)

1 TJU - Tianjin University
2 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique
3 CUHK - The Chinese University of Hong Kong [Hong Kong]

Let $(X _i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $S_n=\sum_{i=1}^n X_i $ and $[ S]_n=\sum_{i=1}^n X_i^2.$ We prove a Cram\'er type moderate deviation expansion for $\mathbf{P}(S_n/\sqrt{[ S]_n} \geq x)$ as $n\to+\infty.$ Our results partly extend the earlier work of Jing, Shao and Wang (Ann. Probab. 31 (2003) 2167–2215) for independent random variables.

Mots clés: Cramér's moderate deviations; Martingales; self-normalized sequences

Domaines: Probabilités [math.PR]

Xiequan Fan, Ion Grama, Quansheng Liu, Qi-Man Shao. Self-normalized Cramér type moderate deviations for martingales. Bernoulli, 2019, 25 (4A), pp.2793-2823. ⟨10.3150/18-BEJ1071⟩. ⟨hal-02487825⟩