Ergodic theorems for dynamic random walks

Abstract

Given any measure-preserving dynamical system ( Y , A , μ , T ) , let g ∈ L p ( μ ) ; we study convergence of the sequence { 1 N ∑ k = 1 N g o T S k , N ≥ 1 } , where S k is a dynamic ℤ r -valued random walk generated by another dynamical system, namely an irrational rotation on the d-dimensional torus. In this note, Van der Corput's inequality, number theory and Gaussian methods are used for studying ergodic theorems and universally representative random sequences.