For a random walk $S_n, n\geq 0$ in $\mathbb{ Z}^d$, let $l(n,x)$ be its local time at the site $x\in \mathbb{ Z}^d$. Define the $\alpha$-fold self intersection local time $L_n(\alpha) := \sum_{ x} l(n,x)^{ \alpha}$, and let $L_n(\alpha|\epsilon, d)$ the corresponding quantity for $d$-dimensional simple random walk. Without imposing any moment conditions, we show that the variances of the local times $\mathop{ var}(L_n(\alpha))$ of any genuinely $d$-dimensional random walk are bounded above by the corresponding characteristics of the simple symmetric random walk in $\mathbb{ Z}^d$, i.e. $\mathop{ var}(L_n(\alpha)) \leq C \mathop{ var}[L_n(\alpha|\epsilon, d)]\sim K_{ d,\alpha}v_{ d,\alpha}(n)$. In particular, variances of local times of all genuinely $d$-dimensional random walks, $d\geq 4$, are similar to the $4$-dimensional symmetric case $\mathop{ var}(L_n(\alpha)) = O(n)$. On the other hand, in dimensions $d\leq 3$ the resemblance to the simple random walk $\liminf_{ n\to \infty} \mathop{ var}(L_n(\alpha))/v_{ d,\alpha}(n)>0$ implies that the jumps must have zero mean and finite second moment.
Categories: math.PR