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

Let ℓ(n,x)ℓ(n,x) be the local time of a random walk on Z2Z2. We prove a strong law of large numbers for the quantity Ln(α)=∑x∈Z2ℓ(n,x)αLn(α)=∑x∈Z2ℓ(n,x)α for all α≥0α≥0. We use this result to describe the distribution of the local time of a typical point in the range of the random walk.