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Let $a_n$ be the random increasing sequence of natural numbers which takeseach 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, forevery 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.

A random pointwise ergodic theorem with Hardy field weights