A note on product sets of random sets

Given two sets of positive integers $A$ and $B$, let $AB := {ab : a \in A,, b \in B}$betheirproductsetandput$A^k := A \cdots A$($k$times$A$) for any positive integer $k$. Moreover, for every positive integer $n$ and every $\alpha \in [0,1]$, let $\mathcal{B}(n,\alpha )$ denote the probabilistic model in which a random set $A\subseteq \{1,\dots ,n\}$ is constructed by choosing independently every element of $\{1,\dots ,n\}$ with probability $\alpha$. We prove that if $A_1,\dots ,A_s$ are random sets in $$\mathcal{B}$(n_1, \alpha_1), \dots, $\mathcal{B}$(n_s, \alpha_s)$,respectively,$k_1, \dots, k_s$ are fixed positive integers, ${\alpha }_i{n}_i\to +\mathrm{\infty }$, and $1\mathrm{/}{\alpha }_i$ does not grow too fast in terms of a product of $\log n_j$; then $|A_1^{k_1} \cdots A_s^{k_s}| \sim \frac{|A_1|^{k_1}}{k_1!}\cdots\frac{|A_s|^{k_s}}{k_s!}$ with probability $1-o(1)$. This is a generalization of a result of Cilleruelo, Ramana, and Ramar'e, who considered the case $s=1$ and $k_1=2$.