Optimal Bounds for the Variance of Self-Intersection Local Times

George Deligiannidis and Sergey Utev

International Journal of Stochastic Analysis, 2016, vol. 2016, 1-10

Abstract:

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.

Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:5370627

DOI: 10.1155/2016/5370627

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