多重收益的几何定律直到危险
Geometric Law for Multiple Returns until a Hazard
摘要：
对于一个$ψ$ -混合平稳过程$ξ_0,ξ_1,ξ_2,...$，我们考虑的次数$\mathcal N_N$的多次复发$\{ξ_{q_i(n)}\inΓ_N,\, i=1,...,\ell\}$到一个集$Γ_N$的$n$，直到时刻$τ_N$（我们称之为一个危险）时，另一个多次复发$\{ξ_{q_i(n)}\inΔ_N,\, i=1,...,\ell\}$发生的第一次，其中$Γ_N\capΔ_N= \emptyset$和$q_i(n)<q_{i+1}(n),\, i=1,...,\ell$是非负增加的功能，采取的整数值的整数。结果表明，如果$P\{ξ_0\inΓ_N\}$和$P\{ξ_0\inΔ_N\}$在$N$中以相同的速度衰减，则$\mathcal N_N$弱收敛于几何分布的随机变量。我们也得到了类似的结果，在动力系统的设置考虑$ψ$ -混合移位$T$序列空间$Ω$和研究的数量多个递归$\{ T^{q_i(n)}ω\in A_n^b,\, i=1,...,\ell\}$，直到第一次出现的另一个多个递归$\{ T^{q_i(n)}ω\in A_m^a,\, i=1,...,\ell\}$其中$A_m^a,\, A_n^b$是圆柱集的长度$m$和$n$序列$a,b\inΩ$，分别构造，并选择，使它们的概率具有相同的顺序。这项工作的动机是一些文件的数量渐近的单次和多次返回到收缩集，以及开放系统的论文研究他们的行为，直到退出通过一个“洞”。
英文摘要：
For a $ψ$-mixing stationary process $ξ_0,ξ_1,ξ_2,...$ we consider the number $\mathcal N_N$ of multiple recurrencies $\{ξ_{q_i(n)}\inΓ_N,\, i=1,...,\ell\}$ to a set $Γ_N$ for $n$ until the moment $τ_N$ (which we call a hazard) when another multiple recurrence $\{ξ_{q_i(n)}\inΔ_N,\, i=1,...,\ell\}$ takes place for the first time where $Γ_N\capΔ_N= \emptyset$ and $q_i(n)<q_{i+1}(n),\, i=1,...,\ell$ are nonnegative increasing functions taking on integer values on integers. It turns out that if $P\{ξ_0\inΓ_N\}$ and $P\{ξ_0\inΔ_N\}$ decay in $N$ with the same speed then $\mathcal N_N$ converges weakly to a geometrically distributed random variable. We obtain also a similar result in the dynamical systems setup considering a $ψ$-mixing shift $T$ on a sequence space $Ω$ and study the number of multiple recurrencies $\{ T^{q_i(n)}ω\in A_n^b,\, i=1,...,\ell\}$ until the first occurence of another multiple recurrence $\{ T^{q_i(n)}ω\in A_m^a,\, i=1,...,\ell\}$ where $A_m^a,\, A_n^b$ are cylinder sets of length $m$ and $n$ constructed by sequences $a,b\inΩ$, respectively, and chosen so that their probabilities have the same order. This work is motivated by a number of papers on asymptotics of numbers of single and multiple returns to shrinking sets, as well as by the papers on open systems studying their behavior until an exit through a "hole".
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
