Cramér's moderate deviations for martingales with applications

Abstract

Let $(ξ_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $X_n=\sum_{i=1}^n ξ_i $ and $ \langle X \rangle_n=\sum_{i=1}^n \mathbf{E}(ξ_i^2|\mathcal{F}_{i-1}).$ We prove Cramér's moderate deviation expansions for $\displaystyle \mathbf{P}(X_n/\sqrt{\langle X\rangle_n} \geq x)$ and $\displaystyle \mathbf{P}(X_n/\sqrt{ \mathbf{E}X_n^2} \geq x)$ as $n\to\infty.$ Our results extend the classical Cramér result to the cases of normalized martingales $X_n/\sqrt{\langle X\rangle_n}$ 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.