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Limit theorems for random Dirichlet series

Author

Listed:
  • Buraczewski, Dariusz
  • Dong, Congzao
  • Iksanov, Alexander
  • Marynych, Alexander

Abstract

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series D(α;z)=∑n≥2(logn)α(ηn+iθn)/nz, properly scaled and normalized, where (ηn,θn)n∈N is a sequence of independent copies of a centered R2-valued random vector (η,θ) with a finite second moment and α>−1/2 is a fixed real parameter. As a consequence, we show that the point processes of complex and real zeros of D(α;z) converge vaguely, thereby obtaining a universality result. In the real case, that is, when P{θ=0}=1, we also prove a law of the iterated logarithm for D(α;z), properly normalized, as z→(1/2)+.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:165:y:2023:i:c:p:246-274
    DOI: 10.1016/j.spa.2023.08.007
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