Joint Poisson distribution of prime factors in sets

Given disjoint subsets T1, ... , Tm 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 T1, ... , Tm 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 T1, ... , Tm are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.