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One dimensional random walks killed on a finite set

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  • Uchiyama, Kôhei

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.

Suggested Citation

  • Uchiyama, Kôhei, 2017. "One dimensional random walks killed on a finite set," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2864-2899.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:9:p:2864-2899
    DOI: 10.1016/j.spa.2017.01.003
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    Cited by:

    1. Uchiyama, Kôhei, 2019. "Asymptotically stable random walks of index 1<α<2 killed on a finite set," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5151-5199.
    2. Kôhei Uchiyama, 2020. "Scaling Limits of Random Walk Bridges Conditioned to Avoid a Finite Set," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1296-1326, September.

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