Randomly perturbed ergodic averages

. 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 (cid:2) max(1 , log(1+ | t | )) dμ f < ∞ via a uniform estimation of trigonometric polynomials. We extend this result to L 2 functions satisfying the weaker condition (cid:2) max(1 , log log(1+ | t | )) dμ f < ∞ . 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.