Moments and distribution of the local times of a transient random walk on \(\mathbb Z^{d}\) (Q1028619)

Let \(\{S_n, n\geq 1\}\) be a transient random walk in \(\mathbb Z^d\) which is genuinely \(d\)-dimensional and \[ l(n,x) = \sum_{i=0}^n {\mathbf 1}_{\{S_i =x\}}, \quad n\in{\mathbb N}, \, x\in\mathbb Z^d \] be its local time process. The authors prove a strong law of large numbers for \[ L_n(\alpha) = \sum_{x\in \mathbb Z^d} l(n,x)^\alpha, \quad n\to \infty. \] Furthermore, they show that the asymptotic law of the local time in a random site uniformly distributed over the range is geometric.