Self-intersections of random walks on lattices

Let {X _n ^d }_n≥0be a uniform symmetric random walk on Z^d, and Π^(d) (a,b)={X _n ^d ∈ Z^d : a ≤ n ≤ b}. Suppose f(n) is an integer-valued function on n and increases to infinity as n↑∞, and let Estimates on the probability of the event are obtained for . As an application, a necessary and sufficient condition to ensure is derived for . These extend some results obtained by Erdős and Taylor about the self-intersections of the simple random walk on Z^d. This revised version was published online in June 2006 with corrections to the Cover Date.