Statistical properties of Markov shifts (part I)
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915548312371200 |
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| author | Hafouta, Yeor |
| author_facet | Hafouta, Yeor |
| contents | We prove central limit theorems, Berry-Esseen type theorems, almost sure invariance principles, large deviations and Livsic type regularity for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j(...,X_{j-1},X_j,X_{j+1},...)$, where $(X_j)$ is an inhomogeneous Markov chain satisfying some mixing assumptions and $f_j$ is a sequence of sufficiently regular functions. Even though the case of non-stationary chains and time dependent functions $f_j$ is more challenging, our results seem to be new already for stationary Markov chains. They also seem to be new for non-stationary Bernoulli shifts (that is when $(X_j)$ are independent but not identically distributed). This paper is the first one in a series of two papers. In \cite{Work} we will prove local limit theorems including developing the related reduction theory in the sense of \cite{DolgHaf LLT, DS}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07757 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistical properties of Markov shifts (part I) Hafouta, Yeor Probability Dynamical Systems Statistics Theory We prove central limit theorems, Berry-Esseen type theorems, almost sure invariance principles, large deviations and Livsic type regularity for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j(...,X_{j-1},X_j,X_{j+1},...)$, where $(X_j)$ is an inhomogeneous Markov chain satisfying some mixing assumptions and $f_j$ is a sequence of sufficiently regular functions. Even though the case of non-stationary chains and time dependent functions $f_j$ is more challenging, our results seem to be new already for stationary Markov chains. They also seem to be new for non-stationary Bernoulli shifts (that is when $(X_j)$ are independent but not identically distributed). This paper is the first one in a series of two papers. In \cite{Work} we will prove local limit theorems including developing the related reduction theory in the sense of \cite{DolgHaf LLT, DS}. |
| title | Statistical properties of Markov shifts (part I) |
| topic | Probability Dynamical Systems Statistics Theory |
| url | https://arxiv.org/abs/2510.07757 |