ABSCISSAS OF THE CONVERGENCE OF MULTIPLE RANDOM DIRICHLET SERIES - Куриляк, Андрій Олегович; Скасків, Олег Богданович; Стасів, Надія Юріївна (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2019)
Let F_ω(s ) = _\|n=0\|^∞ f(n) (ω) {( λ_(n),s)}, where the exponents λ_(n)=(λ^(1)_n_1,…, λ^(p)_n_p)R^p_+=[0,+∞)^p, (n)=(n_1, …, n_p)Z^p_+, Z_+:=N{0}, pN, \|n\|=n_1+…+n_p, and the coefficients коефіцієнти (f_(n)(ω)) are pairwise independent complex random variables. In the paper, particular, there are proved following statements: 1) If τ(λ) : = _\|n\|→+∞ \|n\| /\| λ_(n)\|=0, then in order that Dirichlet series is convergent a.e. in the whole space C^p is necessary and sufficient that (Δ > 0) : _\|n\|=0^+∞ (1-F_(n) ((-Δ\|λ_(n)\|)))<+∞. 2) If τ(λ) = 0 and σG_a(R_+{0})^p м.н., then (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}, xR, (n)Z^+_p, is the distribution function |f_(n)(ω)|, G_a is the set of conjugate abscissas of absolute convergence of the random Dirichlet series F_ω.