  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 a N log N + b N log log N + O(N) where a and b are two
constants depending on eigenvalues of the Markov chain.
