2014/02/13 by Chen, Jeremy
#60C05 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1402.3127
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 mγ, where m is the number of balls in that bin. In this note, we consider the case of γ= 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 γ< 1 the fraction of balls in the "heaviest" bin (the bin with the most balls) tends to 1/d, and for γ> 1 the fraction of balls in the "heaviest" bin tends to 1. To partially fill in the gap for γ= 1, we characterize the limit distribution of the fraction of balls in the "heaviest" bin for γ=1 by providing explicit analytical expressions for all its moments.