Reinforced Random Walks and Random Distributions

Abstract

Consider a classical Polya urn process on a complete binary tree. This process generates an exchangeable sequence of random variables ${Z_n}$, with values in $[0,1]$. It is shown that the empirical distribution $^\# \{ i \leq n:{Z_i} \leq s\} /n$ converges weakly and the distribution of this limit is the same as a standard Dubins-Freedman random distribution. As an application, the variance of the first moment of these Dubins-Freedman distributions is calculated.