Limiting distributions for a class of diminishing urn models

Abstract

In this work we analyze a class of 2 x 2 Pólya-Eggenberger urn models with ball replacement matrix M = (-a 0 \\ c -d), a, d ∈ N and c = pa with p ∈ N₀. We determine limiting distributions by obtaining a precise recursive description of the moments of the considered random variables, which allows us to deduce asymptotic expansions of the moments. In particular, we obtain limiting distributions for the pills problem a = c = d = 1, originally proposed by Knuth and McCarthy. Furthermore, we also obtain limiting distributions for the well-known sampling without replacement urn, a = d = 1 and c = 0, and generalizations of it to arbitrary a, d ∈ N and c = 0. Moreover, we obtain a recursive description of the moment sequence for a generalized problem.