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A.S. convergence for infinite colour Pólya urns associated with stable random walks

Arthur Blanc-Renaudie

Abstract

We answer Problem 11.1 of Janson arXiv:1803.04207 on Pólya urns associated with stable random walk. Our proof use neither martingales nor trees, but an approximation with a differential equation.

A.S. convergence for infinite colour Pólya urns associated with stable random walks

Abstract

We answer Problem 11.1 of Janson arXiv:1803.04207 on Pólya urns associated with stable random walk. Our proof use neither martingales nor trees, but an approximation with a differential equation.
Paper Structure (9 sections, 11 theorems, 48 equations)

This paper contains 9 sections, 11 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Model and motivations
  3. Acknowledgment
  4. Main result and differential equation
  5. Main ideas of the proof
  6. Proof of Proposition \ref{['Main2']}
  7. The bulk : Proof of Lemma \ref{['bulk']}
  8. The tail : Proof of Lemma \ref{['tail']}
  9. Proof of Theorem \ref{['Thm']}

Key Result

Theorem 1

Under eq:CVloi, almost surely, we have the following weak convergence as $n\to \infty$,

Theorems & Definitions (16)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • Lemma 5
  • Lemma 6: Bernstein's inequality
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 6 more