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Moments of Pólya urns balanced in expectation

Colin Desmarais

Abstract

In this work, recent results on the moments of balanced Pólya 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.

Moments of Pólya urns balanced in expectation

Abstract

In this work, recent results on the moments of balanced Pólya 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.
Paper Structure (8 sections, 12 theorems, 95 equations)

This paper contains 8 sections, 12 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Definitions, notation and assumptions
  3. Main Results
  4. Bounding spectral projections of moments
  5. Proof of main results
  6. Applications
  7. Uniform attachment with freezing
  8. Hooking networks

Key Result

Lemma 2.1

JANS:25 Suppose $\lambda_1 = b$ is simple. Then there exists a unique right eigenvector $v_1$ of $A$ such that Furthermore, the projection $P_{\lambda_1}$ is given by $P_{\lambda_1} = v_1 a^T$ and as a consequence for any $v \in \mathbb{C}^q$,

Theorems & Definitions (21)

  • Lemma 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.5
  • Remark 3.6
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 11 more