Almost sure convergence of Polya urn schemes

TL;DR

This paper proves the almost sure convergence of general Polya urn schemes under minimal conditions, providing a method to determine the probability of indefinite continuation based on initial composition.

Contribution

It establishes almost sure convergence for broad Polya urn models with minimal assumptions and introduces a way to compute the probability of indefinite process continuation.

Findings

Almost sure convergence is proven for general Polya urn schemes.

A method to calculate the probability of indefinite continuation is provided.

The convergence result applies under minimal initial composition conditions.

Abstract

For the most general Polya urn schemes, we establish the almost sure convergence of its composition. The only requirement is that there are always enough balls of both colors, so that the extractions can be indefinitely pursued according to the specifications of the model. We also consider the method for determining the probability of fulfilling this requirement, as a function of the initial number of balls of each color.