Friedman vs Pólya

Abstract

Suppose an urn contains initially any number of balls of two colours. One ball is drawn randomly and then put back with $\hat{I}\pm$ balls of the same colour and ${\hat{I}}^2$ balls of the opposite colour. Both cases, ${\hat{I}}^2=0$ and ${\hat{I}}^2>0$ are well known and correspond respectively to Pólya'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 Pólya. Independently of the initial urn composition and independently of $\hat{I}\pm$, ${\hat{I}}^2$, and the value of $p>0$, we show that the proportion of balls of one colour converges almost surely to $\frac{1}{2}$. 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.

4 pages, 1 figure, corrected minor typos