Following Hales (2018), the evolution of Pólya's urn may be interpreted as a walk, a Pólya walk, on the integer lattice $\mathbb{N}^2$. We study the visibility properties of Pólya's walk or, equivalently, the divisibility properties of the composition of the urn. In particular, we are interested in the asymptotic average time that a Pólya walk is visible from the origin, or, alternatively, in the asymptotic proportion of draws so that the resulting composition of the urn is coprime. Via de Finetti's exchangeability theorem, Pólya's walk appears as a mixture of standard random walks. This paper is a follow-up of Cilleruelo-Fernández-Fernández (2019), where similar questions were studied for standard random walks.
@article{10_37236_11424,
author = {Jos\'e L. Fern\'andez and Pablo Fern\'andez},
title = {Some arithmetic properties of {P\'olya's} urn},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {2},
doi = {10.37236/11424},
zbl = {1528.60038},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11424/}
}
TY - JOUR
AU - José L. Fernández
AU - Pablo Fernández
TI - Some arithmetic properties of Pólya's urn
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/11424/
DO - 10.37236/11424
ID - 10_37236_11424
ER -
%0 Journal Article
%A José L. Fernández
%A Pablo Fernández
%T Some arithmetic properties of Pólya's urn
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/11424/
%R 10.37236/11424
%F 10_37236_11424
José L. Fernández; Pablo Fernández. Some arithmetic properties of Pólya's urn. The electronic journal of combinatorics, Tome 30 (2023) no. 2. doi: 10.37236/11424
