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On the Height of One-Dimensional Random Walk

Author

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  • Mohamed Abdelkader

    (Department of Statistics and Operations Research, Faculty of Sciences, King Saud University, Riyadh 11451, Saudi Arabia)

Abstract

Consider the one-dimensional random walk X n : as it evolves (at each unit of time), it either increases by one with probability p or resets to 0 with probability 1 − p . In the present paper, we analyze the law of the height statistics H n , corresponding to our model X n . Also, we prove that the limiting distribution of the walk X n is a shifted geometric distribution with parameter 1 − p and find the closed forms of the mean and the variance of X n using the probability-generating function.

Suggested Citation

  • Mohamed Abdelkader, 2023. "On the Height of One-Dimensional Random Walk," Mathematics, MDPI, vol. 11(21), pages 1-12, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4513-:d:1272525
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    References listed on IDEAS

    as
    1. Endre Csáki & Yueyun Hu, 2001. "Asymptotic Properties of Ranked Heights in Brownian Excursions," Journal of Theoretical Probability, Springer, vol. 14(1), pages 77-96, January.
    2. Kankal, Murat & AkpInar, Adem & Kömürcü, Murat Ihsan & Özsahin, Talat Sükrü, 2011. "Modeling and forecasting of Turkey's energy consumption using socio-economic and demographic variables," Applied Energy, Elsevier, vol. 88(5), pages 1927-1939, May.
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    Keywords

    height; return time; random walk;
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