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From infinite urn schemes to self-similar stable processes

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  • Durieu, Olivier
  • Samorodnitsky, Gennady
  • Wang, Yizao

Abstract

We investigate the randomized Karlin model with parameter β∈(0,1), which is based on an infinite urn scheme. It has been shown before that when the randomization is bounded, the so-called odd-occupancy process scales to a fractional Brownian motion with Hurst index β∕2∈(0,1∕2). We show here that when the randomization is heavy-tailed with index α∈(0,2), then the odd-occupancy process scales to a (β∕α)-self-similar symmetric α-stable process with stationary increments.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:4:p:2471-2487
    DOI: 10.1016/j.spa.2019.07.008
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    References listed on IDEAS

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    1. Lei, Pedro & Nualart, David, 2009. "A decomposition of the bifractional Brownian motion and some applications," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 619-624, March.
    2. Alsmeyer, Gerold & Iksanov, Alexander & Marynych, Alexander, 2017. "Functional limit theorems for the number of occupied boxes in the Bernoulli sieve," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 995-1017.
    3. Pipiras,Vladas & Taqqu,Murad S., 2017. "Long-Range Dependence and Self-Similarity," Cambridge Books, Cambridge University Press, number 9781107039469.
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    Cited by:

    1. Durieu, Olivier & Wang, Yizao, 2022. "Phase transition for extremes of a stochastic model with long-range dependence and multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 55-88.
    2. 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.

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