Randomly Perturbed Ergodic Averages

Convergence properties of random ergodic averages have been extensively studied in the literature. In these notes, we exploit a uniform estimate by Cohen & Cuny who showed convergence of a series along randomly perturbed times for functions in $L^2$ with $\int \max (1,\log (1+\mathrm{\mid }t\mathrm{\mid }))d{\mu }_f<\mathrm{\infty }$. We prove universal pointwise convergence of a class of random averages along randomly perturbed times for $L^2$ functions with $\int \max(1,\log\log(1+|t|)) d\mu_f<\infty$. For averages with additional smoothing properties, we obtain a universal variational inequality as well as universal pointwise convergence of a series define by them for all functions in $L^2$.