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International Journal of Stochastic Analysis
Volume 2016 (2016), Article ID 5370627, 10 pages
Research Article

Optimal Bounds for the Variance of Self-Intersection Local Times

1Department of Statistics, University of Oxford, 24-29 St. Giles, Oxford OX1 3LB, UK
2Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK

Received 22 March 2016; Revised 17 May 2016; Accepted 7 June 2016

Academic Editor: Onesimo Hernandez-Lerma

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.


For a -valued random walk , let be its local time at the site . For , define the -fold self-intersection local time as . Also let be the corresponding quantities for the simple random walk in . Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely -dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, . In particular, for any genuinely -dimensional random walk, with , we have . On the other hand, in dimensions we show that if the behaviour resembles that of simple random walk, in the sense that , then the increments of the random walk must have zero mean and finite second moment.