Finite-sample Borel--Cantelli inequalities under mixing conditions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911623524909056 |
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| author | Panraksa, Chatchawan |
| author_facet | Panraksa, Chatchawan |
| contents | We prove explicit finite-$N$ lower bounds for $\mathbb P(\bigcup_{k=1}^N A_k)$ when the $σ$-algebras generated by an event sequence satisfy quantitative $φ$- or $α$-mixing bounds. The main $φ$-mixing estimate is obtained by a residue-class blocking argument and a one-sided approximate-independence inequality; it has a free spacing parameter $L\ge0$, spacing coefficient $1/(L+1)$, and residual terms governed by $φ(L+1)$. For $α$-mixing families, we derive an additive-correction analogue using strong-mixing covariance control. A windowed rate corollary and a second-order Bonferroni refinement parallel the corresponding $m$-dependent finite-sample results. The coefficient $1/(L+1)$ is sharp as a universal spacing constant only in the zero-residual sense: the full mixing classes contain $L$-dependent block constructions with $φ(L+1)=0$ and $α(L+1)=0$ that asymptotically attain the corresponding bound. This sharpness statement does not assert optimality of the residual penalties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23791 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finite-sample Borel--Cantelli inequalities under mixing conditions Panraksa, Chatchawan Probability 60F15, 60G10, 60E15 We prove explicit finite-$N$ lower bounds for $\mathbb P(\bigcup_{k=1}^N A_k)$ when the $σ$-algebras generated by an event sequence satisfy quantitative $φ$- or $α$-mixing bounds. The main $φ$-mixing estimate is obtained by a residue-class blocking argument and a one-sided approximate-independence inequality; it has a free spacing parameter $L\ge0$, spacing coefficient $1/(L+1)$, and residual terms governed by $φ(L+1)$. For $α$-mixing families, we derive an additive-correction analogue using strong-mixing covariance control. A windowed rate corollary and a second-order Bonferroni refinement parallel the corresponding $m$-dependent finite-sample results. The coefficient $1/(L+1)$ is sharp as a universal spacing constant only in the zero-residual sense: the full mixing classes contain $L$-dependent block constructions with $φ(L+1)=0$ and $α(L+1)=0$ that asymptotically attain the corresponding bound. This sharpness statement does not assert optimality of the residual penalties. |
| title | Finite-sample Borel--Cantelli inequalities under mixing conditions |
| topic | Probability 60F15, 60G10, 60E15 |
| url | https://arxiv.org/abs/2604.23791 |