February 2024 Random walks in cones revisited
Denis Denisov, Vitali Wachtel
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 126-166 (February 2024). DOI: 10.1214/22-AIHP1331


In this paper we continue our study of a multidimensional random walk with zero mean and finite variance killed on leaving a cone. We suggest a new approach that allows one to construct a positive harmonic function in Lipschitz cones under minimal moment conditions. This approach allows also to obtain more accurate information about the behaviour of the harmonic function not far from the boundary of the cone. We also prove limit theorems under new moment conditions.

Dans cet article, nous poursuivons notre étude des marches aléatoires multidimensionnelles ayant dérive nulle, une variance finie, et tuées à la sortie d’un cône. Nous proposons une nouvelle approche, permettant de construire une fonction harmonique positive lorsque le cône possède une régularité Lipschitz et sous des conditions minimales de moments des accroissements de la marche aléatoire. Cette approche permet également de décrire précisément le comportement de la fonction harmonique au voisinage du bord du cône. Nous prouvons finalement des théorèmes limites, sous ces nouvelles hypothèses de moments.

Funding Statement

This research was partially supported by the Ministry of Science and Higher Education of the Russian Federation, agreement 075-15-2019-1620 date 08/11/2019. D. Denisov was supported by a Leverhulme Trust Research Project Grant RPG-2021-105. V. Wachtel was partially supported by DFG.


Dedicated to the memory of Dima Ioffe


The authors gratefully acknowledge hospitality of the Saint-Petersburg Department of Steklov Institute, where a significant part of this work has been done. Special thanks are due to Elena Skripka for the smooth organisation of the visit.


Download Citation

Denis Denisov. Vitali Wachtel. "Random walks in cones revisited." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 126 - 166, February 2024. https://doi.org/10.1214/22-AIHP1331


Received: 19 December 2021; Revised: 29 September 2022; Accepted: 2 October 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718377
Digital Object Identifier: 10.1214/22-AIHP1331

Primary: 60G50
Secondary: 60F17 , 60G40

Keywords: conditioned process , Exit time , Harmonic function , Random walk

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré


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Vol.60 • No. 1 • February 2024
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