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Cramér's moderate deviations for martingales with applications (2204.02562v2)
Published 6 Apr 2022 in math.PR, math.ST, and stat.TH
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_i2|\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 $\displaystyle \mathbf{P}(X_n/\sqrt{ \mathbf{E}X_n2} \geq x)$ 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 $X_n/\sqrt{ \mathbf{E}X_n2}$, with martingale differences satisfying the conditional Bernstein condition. Applications to elephant random walks and autoregressive processes are also discussed.