Open Access
November 2020 Capacity of the range in dimension $5$
Bruno Schapira
Ann. Probab. 48(6): 2988-3040 (November 2020). DOI: 10.1214/20-AOP1442


We prove a central limit theorem for the capacity of the range of a symmetric random walk on $\mathbb{Z}^{5}$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular, an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\ge5$, for the probability that the ranges of two independent random walks intersect. The latter are then used for computing covariances of some intersection events at the leading order.


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Bruno Schapira. "Capacity of the range in dimension $5$." Ann. Probab. 48 (6) 2988 - 3040, November 2020.


Received: 1 August 2019; Revised: 1 April 2020; Published: November 2020
First available in Project Euclid: 20 October 2020

MathSciNet: MR4164459
Digital Object Identifier: 10.1214/20-AOP1442

Primary: 60F05 , 60G50 , 60J45

Keywords: capacity , central limit theorem , intersection of random walk ranges , Random walk , ‎range‎

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 6 • November 2020
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