Randomly perturbed ergodic averages

Abstract:

We consider a class of random ergodic averages, containing averages along random non–integer sequences. For such averages, Cohen & Cuny obtained uniform universal pointwise convergence for functions in $L^2$ with $\int \max (1,\log (1+|t|)) d\mu _f<\infty$ via a uniform estimation of trigonometric polynomials. We extend this result to $L^2$ functions satisfying the weaker condition $\int \max (1,\log \log (1+|t|)) d\mu _f<\infty$. We also prove that uniform universal pointwise convergence in $L^2$ holds for the corresponding smoothed random averages or for random averages whose kernels exhibit sufficient decay at infinity.