A large-deviation result for the range of random walk and for the Wiener sausage |
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Authors: | Yuji Hamana Harry Kesten |
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Institution: | (1) Faculty of Mathematics, Kyushu University 36, Fukuoka 812-8581, Japan. e-mail: hamana@math.kyushu-u.ac.jp, JP;(2) Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY 14853, USA. e-mail: kesten@math.cornell.edu, US |
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Abstract: | Let {S
n
} be a random walk on ℤ
d
and let R
n
be the number of different points among 0, S
1,…, S
n
−1. We prove here that if d≥ 2, then ψ(x) := lim
n
→∞(−:1/n) logP{R
n
≥nx} exists for x≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper.
We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A⊂ℝ
d
let Λ
t
= Λ
t
(A) be the d-dimensional Lebesgue measure of the `sausage' ∪0≤
s
≤
t
(B(s) + A). Then φ(x) := lim
t→∞:
(−1/t) log P{Λ
t
≥tx exists for x≥ 0 and has similar properties as ψ.
Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001 |
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Keywords: | Mathematics Subject Claasification (2000): Primary 60K35 Secondary 82B43 |
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