Precise Tail Behaviour of Some Dirichlet Series

Abstract

AbstractLet $$\eta _1$$ 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 ] .