Decomposable statistics and waiting time in the Markov-Polya urn model

We consider the Markov-Polya urn model. The urn contains a given number of balls of each of N different colours. The balls are sequentially drawn from the urn one at a time, independently of each other and with equal probabilities to be drawn. After each draw the drawn ball is returned into the urn together with c balls of the same colour. The drawing stops when, for the first time, the frequences of k unspecified colours reach or exceed the corresponding (random) levels settled before the beginning of the trials. Limit distributions of the decomposable statistics L-Nk = (j=1)Sigma(N)g(eta(j)) are studied, as N --> infinity and k = k(N), where g is a given function of integer argument and eta(1),..., eta(N) are the frequences, the numbers of balls of the corresponding colours, at the stopping time.