On waiting time in the Markov-Pólya scheme

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 lime v 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 vc(N, k) are derived as consequences of general limit theorems for decomposable statistics L Nk = ∑j=1Ngj(ηj), where g1 . . . . , gN are given functions of integer argument, and η1, . . . , ηN are the frequencies, i.e., the amounts of balls of the corresponding colors, at the stopping time. ©1998 Plenum Publishing Corporation.