The mean number of sites visited by a pinned random walk |
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Authors: | Kôhei Uchiyama |
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Institution: | (1) Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan |
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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.
Supported in part by Monbukagakusho grand-in-aid no. 15540109. |
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Keywords: | Range of random walk Asymptotic expansion Pinned random walk |
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