论文标题
用于矢量半群中亚刚性随机函数的千古定理
An ergodic theorem for subadditive random functions on vector semigroups
论文作者
论文摘要
令$ f =（f^x \ mid x \ in s）$，$ s \ subset \ mathbb {z}^m $，成为概率空间$（ω，\ mathsf {p}）$和$ h $在$ s $上随机函数的概率空间$（ω，\ mathsf {p} $ s $ s $ s $ h（x+h（x x x y y）的实际随机功能， H（x，ω）+H（y，f^xΩ）$ for s $ in s $和$ω\inΩ$的所有$ x，y \。我们证明，存在$ q \ colon o \ to [ - \ infty; \ infty）$上定义了$ o = \ mathrm {int}（\ mathrm {cone}（s））$的$ q \ colon o \ o \ colon o \ o \ colon o \ c \ colon o \ c \ colon o \ colon o = $ q \ colon o \ c（\ mathrm {cone}（s s））$，并且存在$ w \ subSet的完整可能性的完整可能性$ h（ymathrm）对于W $中的所有$ω\，所有序列$（x_n）\子集s $带有渐近方向$ x \ in O $。此刻的时刻条件反映了Semigroup $ f $的大小，而不是$ S $的大小。但是，提出了大约$ h $的额外独立假设。
Let $f=(f^x\mid x\in S)$, $S\subset\mathbb{Z}^m$, be a semigroup of ergodic measure-preserving transformations of a probability space $(Ω,\mathsf{P})$ and $h$ a real random function on $S$, such that $h(x+y,ω)\le h(x,ω)+h(y,f^xω)$ for all $x,y\in S$ and $ω\inΩ$. We prove that there exists a sublinear function $q\colon O\to[-\infty;\infty)$ defined on $O=\mathrm{int}(\mathrm{cone}(S))$, and a set $W\subsetΩ$ of full probability, such that $h(x_n,ω)/\lvert x_n\rvert\to q(x)$ for all $ω\in W$ and all sequences $(x_n)\subset S$ with asymptotic direction $x\in O$. The moment condition for this reflects the size of the semigroup $f$, not that of $S$. However, an additional independence assumption about $h$ is made.
