Mean and variance of balanced Pólya urns
Svante Janson
TL;DR
This paper derives explicit first- and second-moment asymptotics for balanced generalized Pólya urns in the small-urn regime by an elementary martingale-decomposition method. It expresses the urn composition $X_n$ as a sum of orthogonal time-indexed contributions and analyzes these via matrix-function estimates based on the Jordan decomposition of the intensity matrix $A$, yielding precise formulas for $\mathbb{E}X_n$ and $\operatorname{Var}(X_n)$, LLN, and conditions for asymptotic normality. The results cover strictly small and borderline cases, provide a non-degeneracy criterion for the limiting covariance, and illuminate how the long-term behavior depends on the spectral structure, while maintaining a comparatively transparent and direct proof strategy. The findings have practical relevance for applications where mean and variance are of interest and offer a clear framework to compare with central limit theorems for small urns.
Abstract
It is well known that in a small Pólya 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.