ABSCISSAS OF THE CONVERGENCE OF MULTIPLE RANDOM DIRICHLET SERIES

Abstract

Let $F_{\omega}(s ) = \sum_{\|n=0\|}^{\infty} f(n) (\omega) \exp \{( \lambda_{(n)},s)\},$ where the exponents   $\lambda_{(n)}=(\lambda^{(1)}_{n_1},\ldots, \lambda^{(p)}_{n_p})\in\mathbb{R}^p_+=[0,+\infty)^p,$ $(n)=(n_1, \ldots, n_p)\in\mathbb{Z}^p_+,$ $\mathbb{Z}_+:=\mathbb{N}\cup\{0\},$ $p\in\mathbb{N},$ $\|n\|=n_1+\ldots+n_p,$ and the coefficients коефіцієнти $(f_{(n)}(\omega))$  are pairwise independent complex random variables.  In the paper, particular,  there are proved following statements: 1) If $\tau(\lambda) : = \lim_{\|n\|\to+\infty} \ln \|n\| /\| \lambda_{(n)}\|=0,$ then in order that Dirichlet series is convergent a.e. in the whole space $\mathbb{C}^p$ is necessary and sufficient that 
$$(\forall \Delta > 0) : \sum_{\|n\|=0}^{+\infty} (1-F_{(n)} (\exp(-\Delta\|\lambda_{(n)}\|)))<+\infty.$$ 2) If $\tau(\lambda) = 0$ and $\sigma\in\partial G_a\cap(\mathbb{R}_+\setminus\{0\})^p$ м.н., then $$(\forall\varepsilon>0):  \ \sum_{\|n\|=0}^{+\infty} (1-F_{(n)}(e^{(-1+\varepsilon)(\sigma,\lambda_{(n)})}))<+\infty \ \wedge \ \sum_{\|n\|=0}^{+\infty} (1-F_{(n)}(e^{(-1-\varepsilon)(\sigma,\lambda_{(n)})}))=+\infty $$
where $F_{(n)}(x) := P\{\omega:|f_{(n)} (\omega)|<x\},$ $x\in\mathbb{R},$ $(n)\in\mathbb{Z}^+_p,$
is the distribution function $|f_{(n)}(\omega)|,$ $\partial G_a$ is the set of conjugate abscissas of absolute convergence of the random Dirichlet series $F_{\omega}.$