On the Asymptotic moments and Edgeworth expansions for some processes in random dynamical environment

Abstract

We prove that certain asymptotic moments exist for some random distance expanding dynamical systems and Markov chains in random dynamical environment, and compute them in terms of the derivatives at the $0$ of an appropriate pressure function. It will follow that these moments satisfy the relations that the asymptotic moments $\gam_k=\lim_{n\to\infty}n^{-[\frac k2]}\bbE(\sum_{i=1}^n X_i)^k$ of sums of independent and identically distributed random variables satisfy. We will also obtain certain (Edgeworth) asymptotic expansions related to the central limit theorem for such processes. Our proofs rely on a (parametric) random complex Ruelle-Perron-Frobenius theorem, which replaces some the spectral techniques which are used in literature in order to obtain limit theorems for deterministic dynamical systems and Markov chains.