Harmonic measure in a multidimensional gambler's problem

Denis Denisov; Vitali Wachtel

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arXiv · 2212.11526

Harmonic measure in a multidimensional gambler's problem

Denis Denisov·Vitali Wachtel

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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.

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Denis Denisov, Vitali Wachtel. 2022-12-22. Harmonic measure in a multidimensional gambler's problem. https://arxiv.org/abs/2212.11526

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