Waiting times for multiple occurrences of several events

Consider an ‘experiment’ which can be repeated indefinitely often resulting in independent random outcomes. Fix attention on a finite number of possible (sets of) outcomes E1,E2,... and define W=W(N1,N2,...) to be expected number of repetitions needed to ensure that E1 has occurred (at least) N1 times, E2 has occurred (at least) N2 times, etc. This article examines the asymptotic behavior of W as a function of the sum ΣjNj, as the latter grows without bound.