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Crossings States and Sets of States in Random Walks

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  • Vyacheslav M. Abramov

Abstract

We consider a random walk in $$\mathbb {Z}^2$$ Z 2 and establish a number of results for the distributions and expectations of the number of usual (undirected) and specifically defined in the paper directed state-crossings and different sets of states crossings. As well, we extend the results to d-dimensional random walks, $$d\ge 2$$ d ≥ 2 , in bounded areas.

Suggested Citation

  • Vyacheslav M. Abramov, 2023. "Crossings States and Sets of States in Random Walks," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-34, March.
    • Handle: RePEc:spr:metcap:v:25:y:2023:i:1:d:10.1007_s11009-023-09979-0
    DOI: 10.1007/s11009-023-09979-0
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    References listed on IDEAS

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    1. Burq, Zaeem A. & Jones, Owen D., 2008. "Simulation of Brownian motion at first-passage times," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 77(1), pages 64-71.
    2. Robert Jarrow & Philip Protter & A. Sezer, 2007. "Information reduction via level crossings in a credit risk model," Finance and Stochastics, Springer, vol. 11(2), pages 195-212, April.
    3. Ronald W. Wolff, 1982. "Poisson Arrivals See Time Averages," Operations Research, INFORMS, vol. 30(2), pages 223-231, April.
    4. Vyacheslav M. Abramov, 2018. "Conservative and Semiconservative Random Walks: Recurrence and Transience," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1900-1922, September.
    5. P. H. Brill & M. J. M. Posner, 1977. "Level Crossings in Point Processes Applied to Queues: Single-Server Case," Operations Research, INFORMS, vol. 25(4), pages 662-674, August.
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