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

## TL;DR

This paper establishes the existence and computation of asymptotic moments for certain random dynamical systems and Markov chains, and derives Edgeworth expansions, using a novel random complex Ruelle-Perron-Frobenius approach.

## Contribution

It introduces a new method based on a random complex Ruelle-Perron-Frobenius theorem to analyze asymptotic moments and Edgeworth expansions in random dynamical environments.

## Key findings

- Asymptotic moments exist and are computable via derivatives of a pressure function.
- Asymptotic moments satisfy relations similar to those of i.i.d. sums.
- Edgeworth expansions are derived for these processes.

## 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.