We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line {(x 1 , x 2 )|x 1 ≤ 0, x 2 = 0} before time n. It is proved that for aperiodic random walk with mean zero and finite 2 + δ(> 2)-th absolute moment, this probability times n 1/4 converges to some positive constant c * as n → ∞. We show that c * is expressed by using the characteristic function of the increment of the random walk. For the simple random walk, this expression gives c * = 1 + √ 2/(2 (3/4)).
Let H L (ζ,η) be the probability that a two-dimensional simple random walk starting at ξ hits the third quadrant L for the first time at η. The main objective of this paper is to investigate the asymptotic behavior of H L (ζ,η).It is especially proved that there exists a constant C o such that for ξ e Z 2 \L and / e N, \H L (ξ, (-/, 0)) -A L (f, (-/,0)where h L (ξ, •) is the density of the hitting distribution to the third quadrant of two-dimensional standard Brownian motion starting at f. This estimate is sharp at least in the sense that the powers -2/3 and -5/3 can not be improved.
Abstract. This paper establishes a criterion for whether a d-dimensional random walk on the integer lattice Z d visits a space-time subset infinitely often or not. It is a precise analogue of Wiener's test for regularity of a boundary point with respect to the classical Dirichlet problem. The test obtained is applied to strengthen the harder half of Kolmogorov's test for the random walk.
We analyze the Hunter vs. Rabbit game on a graph, which is a model of communication in adhoc mobile networks. Let G be a cycle graph with N nodes. The hunter can move from a vertex to a vertex along an edge. The rabbit can jump from any vertex to any vertex on the graph. We formalize the game using the random walk framework. The strategy of the rabbit is formalized using a one dimensional random walk over Z. We classify strategies using the order O(k −β−1 ) of their Fourier transformation. We investigate lower bounds and upper bounds of the probability that the hunter catches the rabbit. We found a constant lower bound if β ∈ (0, 1) which does not depend on the size N of the graph. We show the order is equivalent to O(1/ log N) if β = 1 and a lower bound is 1/N (β−1)/β if β ∈ (1, 2]. These results help us to choose the parameter β of a rabbit strategy according to the size N of the given graph. We introduce a formalization of strategies using a random walk, theoretical estimation of bounds of a probability that the hunter catches the rabbit, and also show computing simulation results.
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