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Extension of results on generalized Pólya's urns for polynomially self-repelling walks

Elena Kosygina, Laure Marêché, Thomas Mountford, Jonathon Peterson

Abstract

This is a technical note which extends the results of Kosygina, Mountford and Peterson (Ann. Probab., 51(5):1684-1728, 2023, Section 4) about generalized Pólya's urns from a specific weight function $w(n) = (n+1)^{-α}$ to a general family of weight functions satisfying $(w(n))^{-1}=n^α\left(1+2Bn^{-1}+O\left(n^{-2}\right)\right)$ as $n \to \infty$. The latter was considered by Tóth (Ann. Probab., 24(3):1324-1367, 1996) as a part of his study of polynomially self-repelling walks. This extension will be used in forthcoming developments concerning scaling limits of these walks and related processes.

Extension of results on generalized Pólya's urns for polynomially self-repelling walks

Abstract

This is a technical note which extends the results of Kosygina, Mountford and Peterson (Ann. Probab., 51(5):1684-1728, 2023, Section 4) about generalized Pólya's urns from a specific weight function to a general family of weight functions satisfying as . The latter was considered by Tóth (Ann. Probab., 24(3):1324-1367, 1996) as a part of his study of polynomially self-repelling walks. This extension will be used in forthcoming developments concerning scaling limits of these walks and related processes.
Paper Structure (3 sections, 9 theorems, 49 equations)

This paper contains 3 sections, 9 theorems, 49 equations.

Table of Contents

  1. Brief introduction
  2. PSR walks and the associated generalized Pólya urn model
  3. Extension of results about ${\mathfrak{D}}^*$ to the general weight function for PSR walks

Key Result

Lemma 3.1

There exist constants $C_1,c_1>0$ such that for any integers $n,m \geq 1$,

Theorems & Definitions (13)

  • Lemma 3.1: KMP23, Lemma 4.1, Remark 4.2
  • proof
  • Lemma 3.2: KMP23, Lemma 4.3
  • proof
  • Corollary 3.3: KMP23, Corollary 4.4
  • Corollary 3.4: KMP23, Corollary 4.5
  • Lemma 3.5: KMP23, Lemma 4.6
  • proof
  • Lemma 3.6: KMP23, Lemma 4.7
  • Proposition 3.7: KMP23, Proposition 4.8
  • ...and 3 more