Harmonic measure in a multidimensional gambler's problem

TL;DR

This paper analyzes the harmonic measure and Green function behavior in a multidimensional truncated cone, confirming conjectures about gambler's problem asymptotics and providing convergence rates.

Contribution

It introduces asymptotic analysis of harmonic measures in truncated cones and confirms conjectures relating gambler's problem probabilities to Brownian motion approximations.

Findings

Asymptotic behavior of Green function in truncated cones

Confirmation of gambler's problem conjecture on elimination probabilities

Quantitative rate of convergence to Brownian motion approximation

Abstract

We consider a random walk in a truncated cone $K_N$, which is obtained by slicing cone $K$ by a hyperplane at a growing level of order $N$. We study the behaviour of the Green function in this truncated cone as $N$ increases. Using these results we also obtain the asymptotic behaviour of the harmonic measure. The obtained results are applied to a multidimensional gambler's problem studied by Diaconis and Ethier (2022). In particular we confirm their conjecture that the probability of eliminating players in a particular order has the same exact asymptotic behaviour as for the Brownian motion approximation. We also provide a rate of convergence of this probability towards this approximation.