Convergence of a smooth random average and its variation inequality

Abstract

The Ergodic Theorem talks about the convergence of time averages of systems. The pointwise Ergodic Theorem states that the time averages converge to the space averages almost everywhere, for any integrable function f. When we consider in continuous time, because of instrumental limitations, time measurements cannot be taken exactly at any instant of time. Therefore, instead of dealing with averages along arithmetic sequences, in applications one has a smooth average around the time of observation. In this dissertation we investigate the pointwise behavior of smoothed out average with a measure preserving continuous flow on a probability space, K_nf (x) = 1/n∑_{k =0}^n-1 ∫ ϕ _{ϵ_k} (t) f (T_k+tx) dt where ϵ _k are i.i.d positive random variables. We prove a variation inequality for this weighted smoothed average and its convergence a.e. in L² for any realization of the random variable ϵ _k in a set of probability 1.