Cramér's moderate deviations for martingales with applications

Abstract

Let $({\xi }_i,{\mathcal{F}}_i{)}_{i\ge 1}$ be a sequence of martingale differences. Set $X_n={\sum }_{i=1}^n{\xi }_i$ and $⟨X{⟩}_n={\sum }_{i=1}^n\mathbf{E}({\xi }_i^2|{\mathcal{F}}_{i-1}).$ We prove Cramér's moderate deviation expansions for $\mathbf{P}(X_n/\sqrt{⟨X{⟩}_n}\ge x)$ and $\mathbf{P}(X_n/\sqrt{\mathbf{E}{X}_n^2}\ge x)$ as $n\to \infty .$ Our results extend the classical Cramér result to the cases of normalized martingales $X_n/\sqrt{⟨X{⟩}_n}$ and standardized martingales $X_n/\sqrt{\mathbf{E}{X}_n^2}$, with martingale differences satisfying the conditional Bernstein condition. Applications to elephant random walks and autoregressive processes are also discussed.

30 pages

References in corpus (1)

• Exponential inequalities for self-normalized martingales with applications