  Let $a_n$ be the random increasing sequence of natural numbers which takes
each value independently with decreasing probability of order $n^{-\alpha}$, $0
< \alpha < 1/2$. We prove that, almost surely, for every measure-preserving
system $(X,T)$ and every $f \in L^1(X)$ orthogonal to the invariant factor, the
modulated, random averages \[ \sup_{b} \Big| \frac{1}{N} \sum_{n = 1}^N b(n)
T^{a_{n}} f \Big| \] converge to $0$ pointwise almost everywhere, where the
supremum is taken over a set of bounded functions with certain uniform
approximation properties.
