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Almost sure and moment convergence for triangular Pólya urns

Svante Janson

Abstract

We consider triangular Pólya urns and show under very weak conditions a general strong limit theorem of the form $X_{ni}/a_{ni}\to \mathcal{X}_i$ a.s., where $X_{ni}$ is the number of balls of colour $i$ after $n$ draws; the constants $a_{ni}$ are explicit and of the form $n^α\log^γn$; the limit is a.s. positive, and may be either deterministic or random, but is in general unknown. The result extends to urns with subtractions under weak conditions, but a counterexample shows that some conditions are needed. For balanced urns we also prove moment convergence in the main results if the replacements have the corresponding moments. The proofs are based on studying the corresponding continuous-time urn using martingale methods, and showing corresponding results there. We assume for convenience that all replacements have finite second moments.

Almost sure and moment convergence for triangular Pólya urns

Abstract

We consider triangular Pólya urns and show under very weak conditions a general strong limit theorem of the form a.s., where is the number of balls of colour after draws; the constants are explicit and of the form ; the limit is a.s. positive, and may be either deterministic or random, but is in general unknown. The result extends to urns with subtractions under weak conditions, but a counterexample shows that some conditions are needed. For balanced urns we also prove moment convergence in the main results if the replacements have the corresponding moments. The proofs are based on studying the corresponding continuous-time urn using martingale methods, and showing corresponding results there. We assume for convenience that all replacements have finite second moments.
Paper Structure (41 sections, 55 theorems, 333 equations)

This paper contains 41 sections, 55 theorems, 333 equations.

Table of Contents

  1. Introduction
  2. Contents
  3. Some notation and other preliminaries
  4. Standing assumptions
  5. General notation
  6. The colour graph
  7. More notation
  8. Stochastic processes
  9. Continuous-time urn
  10. Analysis of one colour
  11. A colour not influenced by others
  12. A colour only produced by one other colour
  13. The general case for a single colour
  14. The main theorem for continuous time
  15. Proof of the main theorem for discrete time
  16. ...and 26 more sections

Key Result

Theorem 1.8

Let $(X_{ni})_{i\in\mathsf Q}$ be a triangular Pólya urn satisfying the conditions -- below. Then, for every colour $i\in\mathsf Q$, there exists a random variable $\widehat{{\mathcal{X}}}_i$ with $0<\widehat{{\mathcal{X}}}_i<\infty$ a.s such that as ${n\to\infty}$:

Theorems & Definitions (160)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 150 more