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 0 of an appropriate pressure function. It will follow that these moments satisfy the relations that the asymptotic moments $${\gamma }_k=\lim _{n\rightarrow \infty }n^{-[\frac{k}{2}]}{\mathbb {E}}(\sum _{i=1}^n X_i)^k$$ of sums of independent and identically distributed centered random variables satisfy. Under certain mixing conditions we will also estimate the convergence rate towards these limits. The arguments in the proof of these results yield that the partial sums generated by the random Ruelle–Perron–Frobenius triplets and all of their parametric derivatives (considered as functions on the base) corresponding to appropriate random transfer or Markov operators satisfy several probabilistic limit theorems such as the central limit theorem. 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 of the spectral techniques used in literature in order to obtain limit theorems for deterministic dynamical systems and Markov chains.