Joint Poisson distribution of prime factors in sets

Given disjoint subsets $T_1,\dots ,T_m$ of “not too large” primes up to $x$, we establish that for a random integer $n$ drawn from $[1,x]$, the $m$-dimensional vector enumerating the number of prime factors of $n$ from $T_1,\dots ,T_m$ converges to a vector of $m$ independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when $T_1,\dots ,T_m$ are unrestricted, and apply this to the distribution of the number of prime factors from a set $T$ given that $n$ has $k$ total prime factors.