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The mean number of sites visited by a pinned random walk

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Abstract

This paper concerns the number Z n of sites visited up to time n by a random walk S n having zero mean and moving on the d-dimensional square lattice Z d. Asymptotic evaluation of the conditional expectation of Z n given that S 0 = 0 and S n = x is carried out under 2 + δ moment conditions (0 ≤ δ ≤ 2) in the cases d = 2, 3. It gives an explicit form of the leading term and reasonable estimates of the remainder term (depending on δ) valid uniformly in each parabolic region of (x, n). In the case x = 0 the problem has been studied for the simple random walk and its analogue for Brownian motion; the estimates obtained here are finer than or comparable to those found in previous works.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo, 152-8551, Japan

    Kôhei Uchiyama

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  1. Kôhei Uchiyama
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Correspondence to Kôhei Uchiyama.

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Supported in part by Monbukagakusho grand-in-aid no. 15540109.

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Uchiyama, K. The mean number of sites visited by a pinned random walk. Math. Z. 261, 277–295 (2009). https://doi.org/10.1007/s00209-008-0325-6

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  • DOI: https://doi.org/10.1007/s00209-008-0325-6

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