Abstract.
Let {S n } be a random walk on ℤd 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⊂ℝd 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.
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Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001
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Hamana, Y., Kesten, H. A large-deviation result for the range of random walk and for the Wiener sausage. Probab Theory Relat Fields 120, 183–208 (2001). https://doi.org/10.1007/PL00008780
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DOI: https://doi.org/10.1007/PL00008780