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Small counts in nested Karlin’s occupancy scheme generated by discrete Weibull-like distributions

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  • Iksanov, Alexander
  • Kotelnikova, Valeriya

Abstract

A nested Karlin’s occupancy scheme is a symbiosis of classical Karlin’s balls-in-boxes scheme and a weighted branching process. To define it, imagine a deterministic weighted branching process in which weights of the first generation individuals are given by the elements of a discrete probability distribution. For each positive integer j, identify the jth generation individuals with the jth generation boxes. The collection of balls is one and the same for all generations, and each ball starts at the root of the weighted branching process tree and moves along the tree according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights.

Suggested Citation

  • Iksanov, Alexander & Kotelnikova, Valeriya, 2022. "Small counts in nested Karlin’s occupancy scheme generated by discrete Weibull-like distributions," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 283-320.
    • Handle: RePEc:eee:spapps:v:153:y:2022:i:c:p:283-320
    DOI: 10.1016/j.spa.2022.08.006
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    References listed on IDEAS

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    1. Pierpaolo De Blasi & Ramsés H. Mena & Igor Prünster, 2022. "Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(1), pages 143-165, February.
    2. Chebunin, Mikhail & Kovalevskii, Artyom, 2016. "Functional central limit theorems for certain statistics in an infinite urn scheme," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 344-348.
    3. Mikhail Chebunin & Sergei Zuyev, 2022. "Functional Central Limit Theorems for Occupancies and Missing Mass Process in Infinite Urn Models," Journal of Theoretical Probability, Springer, vol. 35(1), pages 1-19, March.
    4. Durieu, Olivier & Samorodnitsky, Gennady & Wang, Yizao, 2020. "From infinite urn schemes to self-similar stable processes," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2471-2487.
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