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Optimal Bounds for the Variance of Self-Intersection Local Times

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posted on 2016-11-09, 10:30 authored by G. Deligiannidis, Sergey Utev
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.

History

Citation

International Journal of Stochastic Analysis, 2016, Article ID 5370627

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

  • VoR (Version of Record)

Published in

International Journal of Stochastic Analysis

Publisher

Hindawi Publishing Corporation

issn

2090-3332

eissn

2090-3340

Acceptance date

2016-06-07

Copyright date

2016

Available date

2016-11-09

Publisher version

https://www.hindawi.com/journals/ijsa/2016/5370627/

Language

en

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