Limiting Distributions for a Class Of Diminishing Urn Models

Abstract

ABSTRACT. In this work we analyze a class of 2 × 2 Pólya-Eggenberger urn models with ball replacement matrix M = −a 0 c −d , a, d ∈ N, and c = p · a with p ∈ N 0 . We obtain limiting distributions for this 2 × 2 urn model 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.

Dynamic increasing returns and innovation diffusion: bringing Polya Urn processes to the empirical data

“…For our purposes here let us consider two types of urn schemes, namely, additions to the urn (Kaniovski and Pflug, 1995;Hill et al, 1980;Arthur et al, 1986Arthur et al, , 1987 and withdrawals from the urn (Panholzer and Kuba, 2012;Kuba and Panholzer, 2012;Siegrist, 1987).…”

On moment sequences and mixed Poisson distributions

“…Here, one is interested in the number of white balls remaining in the urn, after all black balls have been drawn. Several urn models of a similar non-tenable nature have recently received some attention under the name diminishing urn models, see [53] and the references therein.…”

“…It was shown in [53] that the factorial moments of the random variablê X δm,αn = X δm,αn /α are given by…”

“…Using a special case of Theorem 2 it was shown that Y has a discrete distribution. However, the result of [53] contains a small gap: the moments (Γ(1 + αs δ )) s∈N only determine a unique distribution for α/δ ≤ 2, see [32]. Hence, only in this case the (factorial) moments of Y determine a unique distribution.…”

On the area under lattice paths associated with triangular diminishing urn models

“…Recently, a distributional analysis of the parameter "number of white balls remaining in the urn when all black balls are removed starting with m black and n white balls" for the pills problem urn model has been given by Hwang et al [9] using a generating functions approach, leading to a full characterization of the limiting distributions appearing (depending on the growth of m and n). Furthermore, a general study of the behaviour of this parameter for diminishing urn models with a ball replacement matrix given by (1) was conducted in [15]. The sampling without replacement urn is a simple fundamental urn model corresponding to the ball replacement matrix M = −1 0 0 −1 .…”