Moment convergence of balanced P\'olya processes

TL;DR

This paper proves that in balanced small Pólya urn processes, the normalized composition converges in distribution to a normal distribution with all moments converging, providing detailed asymptotics of the moments.

Contribution

It establishes moment convergence for balanced small Pólya urns, extending known distributional convergence results to include all moments.

Findings

Normalized compositions converge to normal distribution

All moments of the process converge to those of the normal distribution

Provides asymptotic behavior of moments in balanced Pólya urns

Abstract

It is known that in an irreducible small P\'olya urn process, the composition of the urn after suitable normalization converges in distribution to a normal distribution. We show that if the urn also is balanced, this normal convergence holds with convergence of all moments, thus giving asymptotics of (central) moments.