Moments and distribution of the local time of a two-dimensional random walk

Let ?(n,x)$?(n,x)$ be the local time of a random walk on Z²${\mathbb{Z}}^2$. We prove a strong law of large numbers for the quantity _Ln(α)=_∑x∈Z2?(n,x)^α$L_n\left(\alpha \right)={\sum }_{x\in {\mathbb{Z}}^2}?{(n,x)}^{\alpha }$ for all α≥0$\alpha \ge 0$. We use this result to describe the distribution of the local time of a typical point in the range of the random walk.