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

Let ℓ ( n , x ) be the local time of a random walk on Z 2 . We prove a strong law of large numbers for the quantity L n ( α ) = ∑ x ∈ Z 2 ℓ ( n , x ) α for all α ≥ 0 . We use this result to describe the distribution of the local time of a typical point in the range of the random walk.