arXiv 2004.022682020-04math.PR

常规和非常规条件下的强Borel-Cantelli性质

The strong Borel--Cantelli property in conventional and nonconventional setups

Yuri Kifer

摘要：

我们研究了强Borel-Cantelli属性的事件和序列空间上的位移，同时考虑了传统和非传统的设置。也就是说，在一定的条件下，事件$Γ_1,Γ_2,...$，我们表明，概率为1 \[ \left(\sum_{n=1}^N\prod_{i=1}^\ell P(Γ_{q_i(n)})\right)^{-1}\sum_{n=1}^N\prod_{i=1}^\ell\mathbb{I}_{Γ_{q_i(n)}}\to 1\,\,\mbox{as}\,\, N\to\infty \]其中$q_i(n),\, i=1,...,\ell$是整数值函数满足一定的假设和$\mathbb{I}_Γ$表示的指标$Γ$。当$\ell=1$（称为常规设置）时，这种收敛可以在$ϕ$混合条件下建立，而当$\ell>1$（称为非常规设置）时，需要更强的$ψ$混合条件.这些结果被推广到序列空间的移位$T$，其中$Γ_{q_i(n)}$被$T^{-q_i(n)}C_n^{(i)}$取代，其中$C_n^{(i)},\, i=1,...,\ell,\, n\geq 1$是圆柱集序列。作为应用，我们研究了某些对数距离函数的最大值和收缩圆柱的（多）击中时间的渐近行为。

英文摘要：

We study the strong Borel-Cantelli property both for events and for shifts on sequence spaces considering both a conventional and a nonconventional setups. Namely, under certain conditions on events $Γ_1,Γ_2,...$ we show that with probability one \[ \left(\sum_{n=1}^N\prod_{i=1}^\ell P(Γ_{q_i(n)})\right)^{-1}\sum_{n=1}^N\prod_{i=1}^\ell\mathbb{I}_{Γ_{q_i(n)}}\to 1\,\,\mbox{as}\,\, N\to\infty \] where $q_i(n),\, i=1,...,\ell$ are integer valued functions satisfying certain assumptions and $\mathbb{I}_Γ$ denotes the indicator of $Γ$. When $\ell=1$ (called the conventional setup) this convergence can be established under $ϕ$-mixing conditions while when $\ell>1$ (called a nonconventional setup) the stronger $ψ$-mixing condition is required. These results are extended to shifts $T$ of sequence spaces where $Γ_{q_i(n)}$ is replaced by $T^{-q_i(n)}C_n^{(i)}$ where $C_n^{(i)},\, i=1,...,\ell,\, n\geq 1$ is a sequence of cylinder sets. As an application we study the asymptotical behavior of maximums of certain logarithmic distance functions and of (multiple) hitting times of shrinking cylinders.

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摘要：

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