Moments of P\'olya urns balanced in expectation

TL;DR

This paper extends the analysis of Pólya urns to unbalanced cases with constant expected change, providing new theoretical results and applications to random trees and networks, including normal limit laws with moment convergence.

Contribution

It generalizes recent Pólya urn moment results to unbalanced urns with constant expected change, and applies these to random trees and network degree distributions.

Findings

Derived moments for unbalanced Pólya urns with constant expected change.

Proved normal limit laws with convergence of all moments for degree distributions.

Applied results to random trees and hooking networks.

Abstract

In this work, recent results on the moments of balanced P\'olya urns are generalized to unbalanced urns, with the condition that the expected change in total activity at each step is constant. We also provide applications of our results to the degree distributions of random trees grown by uniform attachment with freezing and to the degree distribution of hooking networks; in both cases we prove a normal limit law with convergence of all moments.