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 $α$ balls of the same colour and $β$ balls of the opposite colour. Both cases, $β=0$ and $β>0$ 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 $α$, $β$, 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.