Analysis of some exactly solvable diminishing urn models

TL;DR

This paper analyzes exactly solvable diminishing urn models, deriving their distributions and behaviors using generating functions and PDEs, with applications to classical problems like pills, cannibal urns, and OK Corral.

Contribution

It provides exact solutions and limiting distributions for several diminishing urn models using a unified analytical approach.

Findings

Derived exact distributions for various diminishing urns

Established limiting behaviors and distributions

Unified approach using 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.