Asymptotic Gaussianity via coalescence probabilities in the Hammond-Sheffield urn

TL;DR

This paper establishes the asymptotic Gaussianity of sums in the Hammond-Sheffield urn model using coalescence probabilities, functional convergence, and Stein's method, extending previous results and providing new insights into coalescing renewal processes.

Contribution

It proves functional convergence to fractional Brownian motion and offers a new proof of Gaussianity for sums in the Hammond-Sheffield urn, using coalescence probabilities and Stein's method.

Findings

Functional convergence towards fractional Brownian motion.

Asymptotic Gaussianity of renormalised sums in the model.

Control of coalescence probabilities for multiple individuals.

Abstract

For the renormalised sums of the random $\pm 1$-colouring of the connected components of $\mathbb Z$ generated by the coalescing renewal processes in the "power law P\'olya's urn" of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three and four individuals that are randomly sampled from $[n]$. This allows us to obtain a new, conceptual proof of the asymptotic Gaussianity (including the functional convergence) of the renormalised sums of more general colourings, which can be seen as an invariance principle beyond the main result of Hammond and Sheffield. In this proof, a key ingredient of independent interest is a sufficient criterion for the asymptotic Gaussianity of the renormalised sums in randomly coloured random partitions of $[n]$, based on Stein's method. Along the way we also prove a statement on the asymptotics of the coalescence probabilities in the long-range seedbank model of Blath, Gonz\'alez Casanova, Kurt, and Span\`o.