Weighing the "Heaviest" Polya Urn

Abstract

In the classical Polya urn problem, one begins with $d$ bins, each containing one ball. Additional balls arrive one at a time, and the probability that an arriving ball is placed in a given bin is proportional to $m^\gamma$, where $m$ is the number of balls in that bin. In this note, we consider the case of $\gamma = 1$, which corresponds to a process of "proportional preferential attachment" and is a critical point with respect to the limit distribution of the fraction of balls in each bin. It is well known that for $\gamma < 1$ the fraction of balls in the "heaviest" bin (the bin with the most balls) tends to $1/d$, and for $\gamma > 1$ the fraction of balls in the "heaviest" bin tends to $1$. To partially fill in the gap for $\gamma = 1$, we characterize the limit distribution of the fraction of balls in the "heaviest" bin for $\gamma=1$ by providing explicit analytical expressions for all its moments.