Random Walks and Intersection Local Time

With each random walk on $\mathbb{Z}^2$ we associate a functional related to the number of steps which the walk spends in sites occupied at least $k$ times. We show that if our random walk is in the domain of attraction of a stable process of order greater than $2(2k - 2)/(2k - 1)$, then our functional coverges to the intersection local time of the limiting process.