Cramér's moderate deviations for martingales with applications
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913715582926848 |
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| author | Fan, Xiequan Shao, Qi-Man |
| author_facet | Fan, Xiequan Shao, Qi-Man |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_02562 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cramér's moderate deviations for martingales with applications Fan, Xiequan Shao, Qi-Man Probability Statistics Theory 60G42, 60F10, 60E15 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. |
| title | Cramér's moderate deviations for martingales with applications |
| topic | Probability Statistics Theory 60G42, 60F10, 60E15 |
| url | https://arxiv.org/abs/2204.02562 |