Cram\'{e}r's moderate deviations for martingales with applications

TL;DR

This paper extends Cramér's moderate deviation results to normalized and standardized martingales with differences satisfying the Bernstein condition, with applications to random walks and autoregressive models.

Contribution

It provides new moderate deviation expansions for martingales, broadening classical results to more general martingale difference sequences.

Findings

Established Cramér's moderate deviation expansions for normalized martingales.

Extended classical results to martingales with Bernstein condition.

Applied findings to elephant random walks and autoregressive processes.

Abstract

Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $X_n=\sum_{i=1}^n \xi_i $ and $ \langle X \rangle_n=\sum_{i=1}^n \mathbf{E}(\xi_i^2|\mathcal{F}_{i-1}).$ We prove Cram\'er'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 Cram\'{e}r 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.