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Moments of balanced Pólya urns

Svante Janson

TL;DR

The paper develops general $L^p$ moment bounds for the centered composition of balanced Polya urns under broad assumptions, expressible via the eigenstructure of the intensity matrix $A$ with leading eigenvalue $b$. It analyzes a decomposition into deterministic linear evolution plus martingale increments to obtain bounds for $\\|X_n-\\mathbb{E}X_n\\|_p$ that depend on the spectral gap $\\operatorname{Re}\\lambda_2$. Under known asymptotic normality results, these bounds imply convergence of all moments (uniform integrability) for the same limiting distributions. The authors also provide a vector-valued martingale inequality and detailed matrix estimates to support the proofs, and show that the results refine and generalize several earlier findings for balanced urns. The outcomes offer a unified, simpler framework for transferring distributional limits into moment convergence.

Abstract

We give bounds for (central) moments for balanced Pólya urns under very general conditions. In some cases, these bounds imply that moment convergence holds in earlier known results on asymptotic distribution. The results overlap with previously known results, but are here given more generally and with a simpler proof.

Moments of balanced Pólya urns

TL;DR

The paper develops general moment bounds for the centered composition of balanced Polya urns under broad assumptions, expressible via the eigenstructure of the intensity matrix with leading eigenvalue . It analyzes a decomposition into deterministic linear evolution plus martingale increments to obtain bounds for that depend on the spectral gap . Under known asymptotic normality results, these bounds imply convergence of all moments (uniform integrability) for the same limiting distributions. The authors also provide a vector-valued martingale inequality and detailed matrix estimates to support the proofs, and show that the results refine and generalize several earlier findings for balanced urns. The outcomes offer a unified, simpler framework for transferring distributional limits into moment convergence.

Abstract

We give bounds for (central) moments for balanced Pólya urns under very general conditions. In some cases, these bounds imply that moment convergence holds in earlier known results on asymptotic distribution. The results overlap with previously known results, but are here given more generally and with a simpler proof.

Paper Structure

This paper contains 12 sections, 11 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Definition and assumptions
  4. Some notation
  5. The intensity matrix
  6. Eigenvalues and spectral decomposition
  7. Main results
  8. Proofs of main results
  9. A martingale inequality
  10. Matrix estimates
  11. Uniform integrability
  12. Proof of Lemma

Key Result

Lemma 2.2

If the Pólya urn is tenable and balanced, and moreover any colour has a nonzero probability of ever appearing in the urn, then $\operatorname{Re}\lambda\leqslant b$ for every $\lambda\in\sigma(A)$, and, furthermore, if $\operatorname{Re}\lambda=b$ then $\nu_\lambda=0$. We may thus assume $\lambda_1=

Theorems & Definitions (24)

  • Remark 2.1
  • Lemma 2.2: SJ307
  • Lemma 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.5
  • proof : Proof of Theorem
  • proof : Proof of Theorem
  • ...and 14 more