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Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin’s occupancy scheme

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  • Buraczewski, Dariusz
  • Iksanov, Alexander
  • Kotelnikova, Valeriya

Abstract

We prove a law of the iterated logarithm (LIL) for an infinite sum of independent indicators parameterized by t and monotone in t as t→∞. It is shown that if the expectation b and the variance a of the sum are comparable, then the normalization in the LIL includes the iterated logarithm of a. If the expectation grows faster than the variance, while the ratio logb/loga remains bounded, then the normalization in the LIL includes the single logarithm of a (so that the LIL becomes a law of the single logarithm). Applications of our result are given to the number of points of the infinite Ginibre point process in a disk and the number of occupied boxes and related quantities in Karlin’s occupancy scheme.

Suggested Citation

  • Buraczewski, Dariusz & Iksanov, Alexander & Kotelnikova, Valeriya, 2025. "Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin’s occupancy scheme," Stochastic Processes and their Applications, Elsevier, vol. 183(C).
    • Handle: RePEc:eee:spapps:v:183:y:2025:i:c:s0304414925000389
    DOI: 10.1016/j.spa.2025.104597
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    References listed on IDEAS

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    1. 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.
    2. Buraczewski, Dariusz & Dong, Congzao & Iksanov, Alexander & Marynych, Alexander, 2023. "Limit theorems for random Dirichlet series," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 246-274.
    3. Lin Yuan & John Kalbfleisch, 2000. "On the Bessel Distribution and Related Problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 438-447, September.
    4. Stoica, George, 2003. "Functional local law of the iterated logarithm for geometrically weighted random series," Statistics & Probability Letters, Elsevier, vol. 62(1), pages 71-77, March.
    5. Dolgopyat, Dmitry & Hafouta, Yeor, 2022. "Edgeworth expansions for independent bounded integer valued random variables," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 486-532.
    6. Fuqing Gao & Yunshi Gao & Xianjie Xia, 2024. "Asymptotic Behaviors for Random Geometric Series," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2818-2842, September.
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