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February 2021 Some random paths with angle constraints
Clément Berenfeld, Ery Arias-Castro
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Ann. Inst. H. Poincaré Probab. Statist. 57(1): 116-131 (February 2021). DOI: 10.1214/20-AIHP1073


We propose a simple, geometrically-motivated construction of smooth random paths in the plane. The construction is such that, with probability one, the paths have finite curvature everywhere. Our construction is Markovian of order 2. We show that a simpler construction which is Markovian of order 1 fails to exhibit the desired finite curvature property.

Nous étudions une manière élémentaire de construire des marches aléatoires du plan à l’aide d’angles aléatoires. Cette construction, issue de considérations géométriques, est telle que le processus limite possède presque sûrement des trajectoires dont la courbure est partout finie. Les marches aléatoires que nous exhibons sont markoviennes d’ordre 2, et nous montrons qu’une approche plus simple, avec des processus d’ordre 1, ne permet pas d’obtenir, à la limite, les propriétés désirées de courbure finie.


We are grateful to Bruce Driver and Marc Hoffmann for helpful discussions. We are also grateful to a reviewer and associate editor (both anonymous) for their feedback. This work was partially supported by the US National Science Foundation (DMS 1513465, DMS 1916071).


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Clément Berenfeld. Ery Arias-Castro. "Some random paths with angle constraints." Ann. Inst. H. Poincaré Probab. Statist. 57 (1) 116 - 131, February 2021.


Received: 21 November 2018; Revised: 28 April 2020; Accepted: 28 May 2020; Published: February 2021
First available in Project Euclid: 12 March 2021

Digital Object Identifier: 10.1214/20-AIHP1073

Primary: 60F05 , 60G50
Secondary: 60J05

Keywords: Brownian motion , Central limit theorem for dependent variables , curvature , Random angles , Random walk

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré


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Vol.57 • No. 1 • February 2021
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