论文标题
Cramér对申请的中等偏差
Cramér's moderate deviations for martingales with applications
论文作者
论文摘要
令$（ξ_i，\ mathcal {f} _i）_ {i \ geq1} $是一系列martingale差异。设置$ x_n = \ sum_ {i = 1}^nξ_i$和$ \ langle x \ rangle_n = \ sum_ {i = 1}^n \ Mathbf {e}（ξ_i^2 | \ Mathcal \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的结果是标准化的martingales $ x_n/\ sqrt {\ langle x \ rangle_n} $和标准化的martingales $ x_n/\ sqrt {\ mathbf {e} x_n^2} $，具有适合条件的条件差异。还讨论了大象随机步行和自回归过程的应用。
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.
