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 $w(n)=(n+1{)}^{-\alpha }$ to a general family of weight functions satisfying $(w(n){)}^{-1}=n^{\alpha }\left(1+2B{n}^{-1}+O\left(n^{-2}\right)\right)$ 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.

9 pages