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Asymptotically stable random walks of index 1<α<2 killed on a finite set

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  • Uchiyama, Kôhei

Abstract

For a random walk on the integer lattice Z that is attracted to a strictly stable process with index α∈(1,2) we obtain the asymptotic form of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in a natural range of the space and time variables. The situation is relatively simple when the limit stable process has jumps in both positive and negative directions; in the other case when the jumps are one sided rather interesting matters are involved and detailed analyses are necessitated.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:12:p:5151-5199
    DOI: 10.1016/j.spa.2019.02.006
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    References listed on IDEAS

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    1. Uchiyama, Kôhei, 2017. "One dimensional random walks killed on a finite set," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2864-2899.
    2. Vysotsky, Vladislav, 2015. "Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 1886-1910.
    3. Uchiyama, Kôhei, 2011. "A note on summability of ladder heights and the distributions of ladder epochs for random walks," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 1938-1961, September.
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