Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in Zd

For an arbitrary transient random walk ( Sn) n=0 in Zd, d = 1, we prove a strong law of large numbers for the spatial sum similar to x.Zd f (l(n, x)) of a function f of the local times l(n, x) = similar to n i=0 I{Si = x}. Particular cases are the number of (a) visited sites [first considered by Dvoretzky and Erd <spacing diaeresis>os (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) a-fold self-intersections of the random walk [studied by Becker and Konig (J Theor Probab 22:365-374, 2009)], which corresponds to f (i) = ia; (c) sites visited by the random walk exactly j times [considered by Erd <spacing diaeresis>os 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}.