Random walks in Euclidean space

Fix a probability measure on the space of isometries of Euclidean space Rd. Let Y0 = 0, Y1, Y2,... ∈ Rd be a sequence of random points such that Yl+1 is the image of Yl under a random isometry of the previously fixed probability law, which is independent of Yl. We prove a Local Limit Theorem for Yl...

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Bibliographic Details
Published in:Annals of mathematics 2015-01, Vol.181 (1), p.243-301
Main Author: Varjú, Péter Pál
Format: Article
Language:English
Subjects:
Axes of rotation
Central limit theorem
Euclidean space
Fourier transformations
Integers
Lie groups
Mathematical moments
Mathematical theorems
Random walk
Rotation
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Summary:Fix a probability measure on the space of isometries of Euclidean space Rd. Let Y0 = 0, Y1, Y2,... ∈ Rd be a sequence of random points such that Yl+1 is the image of Yl under a random isometry of the previously fixed probability law, which is independent of Yl. We prove a Local Limit Theorem for Yl under necessary nondegeneracy conditions. Moreover, under more restrictive but still general conditions we give a quantitative estimate which describes the behavior of the law of Yl on scales $e^{-cl^{1/4}}\textless r \textless l^{1/2}$.
ISSN:0003-486X
DOI:10.4007/annals.2015.181.1.4