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Weak limits of standardized sums of independent geometrically distributed random variables

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  • Adell, José A.

Abstract

We consider standardized sums of independent geometrically distributed random variables whose failure probabilities approach the unity. We show that such sums converge in law to a random variable having an infinitely divisible distribution whose characteristic function depends on the values of the Riemann zeta function at integer arguments. This is motivated by a probabilistic representation of the Stirling numbers of the second kind.

Suggested Citation

  • Adell, José A., 2025. "Weak limits of standardized sums of independent geometrically distributed random variables," Statistics & Probability Letters, Elsevier, vol. 222(C).
    • Handle: RePEc:eee:stapro:v:222:y:2025:i:c:s0167715225000550
    DOI: 10.1016/j.spl.2025.110410
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    References listed on IDEAS

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    1. Beaulieu Guillaume Boglioni & de Micheaux Pierre Lafaye & Ouimet Frédéric, 2021. "Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin," Dependence Modeling, De Gruyter, vol. 9(1), pages 424-438, January.
    2. Bradley, Richard C. & Pruss, Alexander R., 2009. "A strictly stationary, N-tuplewise independent counterexample to the Central Limit Theorem," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3300-3318, October.
    3. José Antonio Adell & Alberto Lekuona, 2007. "Berry–Esseen Bounds for Standardized Subordinators via Moduli of Smoothness," Journal of Theoretical Probability, Springer, vol. 20(2), pages 221-235, June.
    Full references (including those not matched with items on IDEAS)

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    2. Beaulieu Guillaume Boglioni & de Micheaux Pierre Lafaye & Ouimet Frédéric, 2021. "Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin," Dependence Modeling, De Gruyter, vol. 9(1), pages 424-438, January.
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