In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $\nu $ . We consider a class of non-square-integrable observables $\phi $ , mostly of form $\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$ , where $x_0$ is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index $\alpha \in (0,2)$ . The two types of maps we concatenate are a class of piecewise $C^2$ expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and $\alpha $ , we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by $\nu $ . This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.

中文翻译：

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随机动力系统的稳定定律

在本文中，我们考虑通过连接作用于单位区间的映射形成的随机动力系统 $[0,1]$ 以独立同分布（iid）的方式。随机动力系统被视为平稳马尔可夫过程，具有独特的平稳测度 $\n$ 。我们考虑一类不可平方可积的可观测量 $\phi$ ，大部分是形式 $\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$ ， 在哪里 $x_0$ 是一个非经常性点（特别是非周期点），满足一些其他通用条件，更一般地，具有指数的规则变化的可观测量 $\alpha \in (0,2)$ 。我们连接的两种类型的映射是一类分段 $C^2$ 扩展地图和一类在原点具有无关固定点的间歇地图。在动力学条件下 $\阿尔法$ ，我们建立了泊松极限定律、缩放伯克霍夫和收敛到稳定极限定律以及退火和淬火情况下的函数稳定极限定律。几乎每个淬火实现的极限定律的比例常数与退火情况的比例常数相同，并由下式确定： $\n$ 。这与淬火中心极限定理中的缩放形成对比，其中中心常数以关键方式取决于实现，并且几乎每个实现都不相同。