Randomly perturbed ergodic averages

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