Precise Tail Behaviour of Some Dirichlet Series

Alexander Iksanov () and Vitali Wachtel ()
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Alexander Iksanov: Taras Shevchenko National University of Kyiv
Vitali Wachtel: University of Bielefeld

Journal of Theoretical Probability, 2024, vol. 37, issue 3, 2704-2737

Abstract: Abstract Let $$\eta _1$$ η 1 , $$\eta _2,\ldots $$ η 2 , … be independent copies of a random variable $$\eta $$ η with zero mean and finite variance which is bounded from the right, that is, $$\eta \le b$$ η ≤ b almost surely for some $$b>0$$ b > 0 . Considering different types of the asymptotic behaviour of the probability $$\mathbb {P}\{\eta \in [b-x,b]\}$$ P { η ∈ [ b - x , b ] } as $$x\rightarrow 0+$$ x → 0 + , we derive precise tail asymptotics of the random Dirichlet series $$\sum _{k\ge 1}k^{-\alpha }\eta _k$$ ∑ k ≥ 1 k - α η k for $$\alpha \in (1/2, 1]$$ α ∈ ( 1 / 2 , 1 ] .

Keywords: Change of measure; Local limit theorem; Precise tail behaviour; Random Dirichlet series; Primary: 60G50; 60E99; Secondary: 60F05 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-024-01318-4

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