One dimensional random walks killed on a finite set

Kôhei Uchiyama

Stochastic Processes and their Applications, 2017, vol. 127, issue 9, 2864-2899

Abstract: We study the transition probability, say pAn(x,y), of a one-dimensional random walk on the integer lattice killed when entering into a non-empty finite set A. The random walk is assumed to be irreducible and have zero mean and a finite variance σ2. We show that pAn(x,y) behaves like [gA+(x)ĝA+(y)+gA−(x)ĝA−(y)](σ2/2n)pn(y−x) uniformly in the regime characterized by the conditions ∣x∣∨∣y∣=O(n) and ∣x∣∧∣y∣=o(n) generally if xy>0 and under a mild additional assumption about the walk if xy<0. Here pn(y−x) is the transition kernel of the random walk (without killing); gA± are the Green functions for the ‘exterior’ of A with ‘pole at ±∞’ normalized so that gA±(x)∼2∣x∣/σ2 as x→±∞; and ĝA± are the corresponding Green functions for the time-reversed walk.

Keywords: Exterior domain; Transition probability; Escape from a finite set; Hitting probability of a finite set; Potential theory (search for similar items in EconPapers)
Date: 2017
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spa.2017.01.003

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