Moderate Deviation Principle for a Stochastic Approximation Process
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910200735203328 |
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| author | Shi, Jianan Yin, Qing Miao, Yu |
| author_facet | Shi, Jianan Yin, Qing Miao, Yu |
| contents | In this paper, we investigate a stochastic approximation procedure $\left(X_n\right)_{n\ge 0}$ taking values in $R$. The process is adapted to a filtration $(F_n)_{n\ge 0}$ and satisfies the recursion $X_{n+1}=X_n+\frac{b}{n+1}\big[g(X_n)+U_{n+1}\big]$, where $b>0$, $g:R \to R$ is a function and $\left(U_n\right)_{n\ge 1}$ is a sequence of bounded martingale differences adapted to the filtration $(F_n)_{n\ge 1}$. We establish the moderate deviation principle for the stochastic process $(X_n)_{n\ge 0}$. As auxiliary results, we also obtain the exponential inequality for $(X_n)_{n\ge 0}$ and the moderate deviation principle for weighted sums of bounded martingale differences. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_07369 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Moderate Deviation Principle for a Stochastic Approximation Process Shi, Jianan Yin, Qing Miao, Yu Probability In this paper, we investigate a stochastic approximation procedure $\left(X_n\right)_{n\ge 0}$ taking values in $R$. The process is adapted to a filtration $(F_n)_{n\ge 0}$ and satisfies the recursion $X_{n+1}=X_n+\frac{b}{n+1}\big[g(X_n)+U_{n+1}\big]$, where $b>0$, $g:R \to R$ is a function and $\left(U_n\right)_{n\ge 1}$ is a sequence of bounded martingale differences adapted to the filtration $(F_n)_{n\ge 1}$. We establish the moderate deviation principle for the stochastic process $(X_n)_{n\ge 0}$. As auxiliary results, we also obtain the exponential inequality for $(X_n)_{n\ge 0}$ and the moderate deviation principle for weighted sums of bounded martingale differences. |
| title | Moderate Deviation Principle for a Stochastic Approximation Process |
| topic | Probability |
| url | https://arxiv.org/abs/2605.07369 |