Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$

Asymont, Inna and Korshunov, Dmitry (2020) Strong law of large numbers for a function of the local times of a transient random walk in $Z^d$. Journal of Theoretical Probability, 33. 2315–2336. ISSN 0894-9840

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Abstract

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 ∑x∈Zd f(l(n,x)) of a function f of the local times l(n,x)=∑ni=0 II{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)= II{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)=iα; (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)=II{i=j}.