Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in Z^d

For an arbitrary transient random walk (Sn)n≥0 in Z^d, d≥ 1 , we prove a strong law of large numbers for the spatial sum ∑x∈Zdf(l(n,x)) of a function f of the local times l(n,x)=∑i=0nI{Si=x}. Particular cases are the number of(a)visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function f(i) = I{ i≥ 1 } ;(b)α-fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to f(i) = i^α;(c)sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where f(i) = I{ i= j}.