  Let $a_n$ be the random increasing sequence of natural numbers which takes
each value independently with probability $n^{-a}$, $0 < a < 1/2$, and let
$p(n) = n^{1+\epsilon}$, $0 < \epsilon < 1$. We prove that, almost surely, for
every measure-preserving system $(X,T)$ and every $f \in L^1(X)$ the modulated,
random averages \[ \frac{1}{N} \sum_{n = 1}^N e(p(n)) T^{a_n(\omega)} f\]
converge to $0$ pointwise almost everywhere.
