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October 2021 Remarks on the range and multiple range of a random walk up to the time of exit
Thomas Doehrman, Sunder Sethuraman, Shankar C. Venkataramani
Rocky Mountain J. Math. 51(5): 1603-1614 (October 2021). DOI: 10.1216/rmj.2021.51.1603


We consider the scaling behavior of the range and p-multiple range, that is the number of points visited and the number of points visited exactly p1 times, of a simple random walk on d, for dimensions d2, up to time of exit from a domain DN of the form DN=ND, where Dd, as N. Recent papers have discussed connections of the range and related statistics with the Gaussian free field, identifying in particular that the distributional scaling limit for the range, in the case D is a cube in d3, is proportional to the exit time of Brownian motion. The purpose of this note is to give a concise, different argument that the scaled range and multiple range, in a general setting in d2, both weakly converge to proportional exit times of Brownian motion from D, and that the corresponding limit moments are “polyharmonic”, solving a hierarchy of Poisson equations.


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Thomas Doehrman. Sunder Sethuraman. Shankar C. Venkataramani. "Remarks on the range and multiple range of a random walk up to the time of exit." Rocky Mountain J. Math. 51 (5) 1603 - 1614, October 2021.


Received: 2 September 2020; Revised: 18 January 2021; Accepted: 16 February 2021; Published: October 2021
First available in Project Euclid: 17 February 2022

Digital Object Identifier: 10.1216/rmj.2021.51.1603

Primary: 60F05 , 60G50

Keywords: Brownian motion , constrained , exit , multiple , polyharmonic , Random walk , ‎range‎ , time

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium


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Vol.51 • No. 5 • October 2021
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