Mean and variance of balanced P\'olya urns

TL;DR

This paper analyzes the asymptotic behavior of the mean and covariance in balanced Pólya urns, showing their convergence to the parameters of the limiting normal distribution, extending known results for small urns.

Contribution

It provides explicit asymptotic formulas for the mean and covariance matrix in balanced Pólya urns, generalizing previous results beyond small urns.

Findings

Mean and covariance asymptotics derived

Normalized mean and covariance converge to normal distribution parameters

Extends known results to balanced urns with eigenvalue conditions

Abstract

It is well known that in a small P\'olya urn, i.e., an urn where second largest real part of an eigenvalue is at most half the largest eigenvalue, the distribution of the numbers of balls of different colours in the urn is asymptotically normal under weak additional conditions. We consider the balanced case, and then give asymptotics of the mean and the covariance matrix, showing that after appropriate normalization, the mean and covariance matrix converge to the mean and variance of the limiting normal distribution.