Limit theorems for random Dirichlet series

Dariusz Buraczewski, Congzao Dong, Alexander Iksanov and Alexander Marynych

Stochastic Processes and their Applications, 2023, vol. 165, issue C, 246-274

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)+.

Keywords: Functional central limit theorem; Law of the iterated logarithm; Random Dirichlet series; Space of analytic functions (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2023.08.007

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