Friedman vs P\'olya

TL;DR

This paper analyzes a mixed urn model combining Pólya's and Friedman's schemes, proving that the proportion of one color converges almost surely to 1/2 regardless of initial conditions or parameters.

Contribution

It introduces and studies a new mixed urn model blending Pólya's and Friedman's replacement schemes, establishing almost sure convergence to an equal proportion.

Findings

Proportion of one color converges to 1/2 almost surely.

Convergence holds for any initial urn composition and parameters.

The result generalizes known schemes by combining them with a probabilistic mixture.

Abstract

Suppose an urn contains initially any number of balls of two colours. One ball is drawn randomly and then put back with $\alpha$ balls of the same colour and $\beta$ balls of the opposite colour. Both cases, $\beta=0$ and $\beta>0$ are well known and correspond respectively to P\'olya's and Friedman's replacement schemes. We consider a mixture of both of these: with probability $p\in(0,1]$ balls are replaced according to Friedman's recipe and with probability $1-p$ according to the one by P\'olya. Independently of the initial urn composition and independently of $\alpha$, $\beta$, and the value of $p>0$, we show that the proportion of balls of one colour converges almost surely to $\frac12$. The latter is the limit behaviour obtained by using Friedman's scheme alone, i.e. when $p=1$. Our result follows by adapting an argument due to D. S. Ornstein.