  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.
