Optimal Bounds for the Variance of Self-Intersection Local Times

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.