Random Sequences and Pointwise Convergence of Multiple Ergodic Averages

We prove pointwise convergence, as N -> infinity, for the multiple ergodic averages (1/N) Sigma(N)(n=1) f(T(n)x) . g(S(an)x), where T and S are commuting measure preserving transformations, and a(n) is a random version of the sequence [n(c)] for some appropriate c > 1. We also prove similar mean convergence results for averages of the form (1/N) Sigma(N)(n=1) f(T(an)x) . g(S(an)x), as well as pointwise results when T and S are powers of the same transformations. The deterministic versions of these results, where one replaces a(n) with [n(c)], remain open, and we hope that our method will indicate a fruitful way to approach these problems as well.