Computing hitting times via fluid approximation: application to the coupon collector problem

In this paper, we show how to use stochastic approximation to compute hitting time of a stochastic process, based on the study of the time for a fluid approximation of this process to be at distance 1/N of its fixed point. This approach is developed to study a generalized version of the coupon collector problem. The system is composed by N independent identical Markov chains. At each time step, one Markov chain is picked at random and performs one transition. We show that the time at which all chains have hit the same state is bounded by c1N logN + c2N log logN + O(N) where c1 and c2 are two constants depending on eigenvalues of the Markov chain.