Optimal Bounds for the Variance of Self-Intersection Local Times

George Deligiannidis, Sergey Utev
2016 International Journal of Stochastic Analysis  
For aZd-valued random walk(Sn)n∈N0, letl(n,x)be its local time at the sitex∈Zd. Forα∈N, define theα-fold self-intersection local time asLn(α)≔∑xl(n,x)α. Also letLnSRW(α)be the corresponding quantities for the simple random walk inZd. Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinelyd-dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that
more » ... In particular, for any genuinelyd-dimensional random walk, withd≥4, we havevar⁡(Ln(α))=O(n). On the other hand, in dimensionsd≤3we show that if the behaviour resembles that of simple random walk, in the sense thatlim infn→∞var⁡Lnα/var⁡(LnSRW(α))>0, then the increments of the random walk must have zero mean and finite second moment.
doi:10.1155/2016/5370627 fatcat:mfgejeukdfcfpfymrsqfgnfu2a