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Title: Optimal Bounds for the Variance of Self-Intersection Local Times
Authors: Deligiannidis, G.
Utev, Sergey
First Published: 2016
Publisher: Hindawi Publishing Corporation
Citation: International Journal of Stochastic Analysis, 2016, Article ID 5370627
Abstract: For a Zd-valued random walk (Sn)n N0, let l(n,x) be its local time at the site x Zd. For α N, define the α-fold self-intersection local time as Ln(α) xl(n,x)α. Also let LnSRW(α) be the corresponding quantities for the simple random walk in Zd. Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely d-dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, var(Ln(α))=O(var (LnSRW(α))). In particular, for any genuinely d-dimensional random walk, with d≥4, we have var (Ln(α))=O(n). On the other hand, in dimensions d≤3 we show that if the behaviour resembles that of simple random walk, in the sense that lim infn→∞var Lnα/var(LnSRW(α))>0, then the increments of the random walk must have zero mean and finite second moment.
DOI Link: 10.1155/2016/5370627
ISSN: 2090-3332
eISSN: 2090-3340
Version: Publisher Version
Status: Peer-reviewed
Type: Journal Article
Rights: Copyright © 2016 George Deligiannidis and Sergey Utev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Appears in Collections:Published Articles, Dept. of Mathematics

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