On convergence of random multiple Dirichlet series
Journal Title: Математичні Студії - Year 2018, Vol 49, Issue 2
Abstract
Let Fω(s)=∑∞∥n∥=0f(n)(ω)exp{(λ(n),s)},\ where the exponents λ(n)=(λ(1)n1,…,λ(p)np)∈Rp+, (n)=(n1,…,np)∈Zp+, p∈N, ∥n∥=n1+…+np, and the coefficients f(n)(ω) are pairwise independent random complex variables. In the paper, in particular, we prove the following statements: 1) If τ(λ)=lim¯¯¯¯¯¯¯∥n∥→+∞ln∥n∥/∥λ(n)∥=0, then in order that a Dirichlet series be convergent a.s. in the whole space Cp, it is necessary and sufficient that (∀Δ>0): ∑+∞∥n∥=0(1−F(n)(exp(−Δ∥λ(n)∥)))≤+∞. 2) If τ(λ)=0, then in order that σ∈∂Ga∩(R+∖{0})p a.s., it is necessary and sufficient that (∀ε>0): ∑∥n∥=0+∞(1−F(n)(e(−1+ε)(σ,λ(n))))≤+∞ ∧ ∑∥n∥=0+∞(1−F(n)(e(−1−ε)(σ,λ(n))))=+∞, where F(n)(x):=P{ω:|f(n)(ω)|≤x}, x∈R, (n)∈Zp+ is the distribution function of |f(n)(ω)|, ∂Ga is the set of conjugate abscissas of absolute convergence of the random Dirichlet series Fω.
Authors and Affiliations
Andriy Kuryliak, Oleh Skaskiv, N. Yu. Stasiv
Keywords
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- EP ID EP411719
- DOI 10.15330/ms.49.2.122-137
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