2025/02/24 by Alves, Raphael, Rosales, Rafael A.
#60J05 #FOS: Mathematics #Primary: 60F15 #Probability (math.PR) #Secondary: 60G42
paper · doi:10.48550/arxiv.2502.16768
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∈(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.