Extension of results on generalized P\'olya's urns for polynomially self-repelling walks

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 P\'olya's urns from a specific weight function $w(n) = (n+1)^{-\alpha}$ to a general family of weight functions satisfying $(w(n))^{-1}=n^{\alpha}\left(1+2Bn^{-1}+O\left(n^{-2}\right)\right)$ as $n \to \infty$. The latter was considered by T\'oth (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.