# On waiting time in the Markov-Pólya scheme

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## Abstract

We consider the Markov-Pólya urn scheme. The urn contains a given number of balls of each of N different colors. 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 ball drawn is returned into the urn together with c balls of the same color, c∈{−1, 0, 1, 2, ...}. The drawing process halts when, for the first time, the frequencies of k unspecified colors attain or exceed the corresponding (random) levels settled before the beginning of the trials. Limit distributions of the stopping time υ where g

_{c}(N, K) are considered as N→∞, k=k(N); in particular, the dependence of the asymptotic properties of the waiting time on the parameter c is studied. Results concerning υ_{c}(N, K) are derived as consequences of general limit theorems for decomposable statistics$$L_{NK} = \sum\limits_{j = 1}^N {g_j (\eta _j )} $$

_{1}, ..., g_{N}are given functions of integer argument, and η_{1}...η_{N}are the frequencies, i.e., the amounts of balls of the corresponding colors, at the stopping time.## Keywords

Limit Theorem Limit Distribution Birth Process Random Level Integer Argument
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