Moments and Distribution of the Local Times of a Transient Random Walk on ℤ d

Mathias Becker () and Wolfgang König ()
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Mathias Becker: Universität Leipzig
Wolfgang König: Universität Leipzig

Journal of Theoretical Probability, 2009, vol. 22, issue 2, 365-374

Abstract: Abstract Consider an arbitrary transient random walk on ℤ d with d∈ℕ. Pick α∈[0,∞), and let L n (α) be the spatial sum of the αth power of the n-step local times of the walk. Hence, L n (0) is the range, L n (1)=n+1, and for integers α, L n (α) is the number of the α-fold self-intersections of the walk. We prove a strong law of large numbers for L n (α) as n→∞. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by Černý (Stoch. Proc. Appl. 117:262–270, 2007). Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.

Keywords: Random walk on ℤ d; Local time; Self-intersection number; 60G50; 60J55; 60F15 (search for similar items in EconPapers)
Date: 2009
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-008-0168-4

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