[Paper Review] Analysis of some exactly solvable diminishing urn models

Hsien‐Kuei Hwang, Markus Kuba+1 more|arXiv (Cornell University)|Dec 9, 2022

Stochastic processes and statistical mechanics13 references22 citations

TL;DR

The paper derives exact generating-function solutions and limiting distributions for several diminishing Pólya–Eggenberger urns (pills, cannibal urn, OK Corral) using recurrences solved by generating functions and PDEs.

ABSTRACT

We study several exactly solvable Polya-Eggenberger urn models with a \emph{diminishing} character, namely, balls of a specified color, say $x$ are completely drawn after a finite number of draws. The main quantity of interest here is the number of balls left when balls of color $x$ are completely removed. We consider several diminishing urns studied previously in the literature such as the pills problem, the cannibal urns and the OK Corral problem, and derive exact and limiting distributions. Our approach is based on solving recurrences via generating functions and partial differential equations.

Motivation & Objective

• Motivate study of diminishing urn models where certain color balls are removed after finite steps.
• Develop exact generating-function and PDE methods to solve two-color diminishing urn recurrences.
• Obtain both exact distribution formulas and asymptotic limiting distributions for key models (pills, cannibal urn, OK Corral).
• Show how these methods extend to multi-size pill generalizations and related variants.

Proposed method

• Model the urn dynamics with two colors via a transition matrix M and absorbing states S.
• Translate the evolution into probability generating function recurrences with boundary conditions (Eq. 2a, 2b).
• Solve resulting first-order linear PDEs for the generating functions using the method of characteristics.
• Normalize boundary terms to obtain solvable PDEs in Type A cases and use analytic transforms (η,ξ) to express solutions.
• Extract coefficients to obtain explicit forms for probabilities and moments (Eq. 11, 12, etc.).
• Derive limiting distributions through factorial moments and Beta-function asymptotics for various regimes.

Experimental results

Research questions

• RQ1 What are the probabilities of reaching absorbing states in diminishing urn models starting from given (m,n)?
• RQ2 What is the distribution of the number of remaining white (or black) balls when absorption occurs?
• RQ3 Can exact generating-function solutions be obtained for pills-type, cannibal, and OK Corral diminishing urns?
• RQ4 What are the_limiting distributions under different growth regimes of parameters (e.g., m→∞, n fixed, etc.)?

Key findings

• Derive explicit generating-function formulas for several diminishing urns (pills, cannibal, OK Corral).
• Pills problem: X_{n,m} has generating function h_{n,m}(v)=m v ∫_0^1 (1+(v-1)q)^n (1-q-(v-1)q log q)^{m-1} dq.
• Limiting results: when m→∞ with n,m→∞ suitably, X_{n,m} scaled converges to an exponential distribution; when n→∞ with m fixed, X_{n,m}/n converges to Beta(1,m).
• Variant treatments yield Rayleigh and Beta(1,m) square-root distributions in the m→∞ or n→∞ regimes for other models.
• Cannibal urn results provide an explicit probability: P{X_{n,m}=k} with a closed form series and show asymptotic normality when variance grows; local and Poisson limits discussed as future work.
• OK Corral gives a closed-form survival probability p_{n,m} for all black balls removed, matching known results; multi-parameter extensions discussed.