  We prove pointwise convergence, as $N\to \infty$, for the multiple ergodic
averages $\frac{1}{N}\sum_{n=1}^N f(T^nx)\cdot g(S^{a_n}x)$, 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 $\frac{1}{N}\sum_{n=1}^N
f(T^{a_n}x)\cdot g(S^{a_n}x)$, as well as pointwise results when $T$ and $S$
are powers of the same transformation. 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.
