论文标题
强大的Borel-Cantelli属性在常规和非惯例设置中
The strong Borel--Cantelli property in conventional and nonconventional setups
论文作者
论文摘要
我们研究了考虑常规设置和非常规设置的序列空间的事件和偏移的强大borel-cantelli特性。即，在某些情况下，在事件的某些条件下$γ_1，γ_2，... $我们以概率为一\ [\ left（\ sum_ {n = 1}^n \ prod_ p（γ_{q_i（n）}）\ right）^{ - 1} \ sum_ {n = 1}^n \ prod_ {i = 1}^\ ell \ ell \ mathbb {i} _ {γ_{γ_{q_i（q_i（q_i）{q_i（n）}}}}}} \ to 1 \ to 1 \ to 1 \ \ \ \ mbox \ c \ arbober \ arbober \ arbober \ ar \ ar \ ar \ \ ar \ \ ar \ \ \ \ \ \ \ \ \ \ \ c \ \ \ ar \ \ ar \ \ ar \ ,, \]其中$ q_i（n），\，i = 1，...，\ ell $是满足某些假设的整数值函数，而$ \ mathbb {i}_γ$表示$γ$的指示器。当$ \ ell = 1 $（称为常规设置）时，可以在$ ϕ $混合条件下建立此收敛性，而当$ \ ell> 1 $（称为非惯例设置）时，则需要更强的$ψ$混合条件。这些结果扩展到序列空间的$ t $，其中$γ_{q_i（n）} $由$ t^{ - q_i（n）} c_n^{（i）} $替换，其中$ c_n^{（i）}其中作为应用程序，我们研究了某些对数距离函数的最大值和（多个）击中圆柱体的（多个）击球时间的渐近行为。
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.
