Stable laws for random dynamical systems

TL;DR

This paper studies the statistical behavior of random dynamical systems on the interval, establishing stable limit laws for certain observables and showing that quenched and annealed scalings coincide under specific conditions.

Contribution

It proves Poisson and stable limit laws for non-square-integrable observables in random dynamical systems, highlighting the universality of scaling constants across realizations.

Findings

Poisson limit laws established for certain observables.

Convergence to stable laws for Birkhoff sums.

Quenched and annealed scalings coincide for almost every realization.

Abstract

In this paper we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an iid 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 $\phi(x)=d(x,x_0)^{-\frac{1}{\alpha}}$ where $x_0$ is 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.