Abstract:

By studying the convergence and the growth of bi-random Dirichlet series ∑^∞_n=0a_nX_n(ω)e^-λ_n(ω)s in with coefficients {X_n}(ω): n≥0, being unidentically distributed B-valued φ-mixing random sequences, satisfying d ^p σ_n^p \triangleq d ^pE ll X_nll^p≤ E^pll X_nll< ∞ (p>1, d>0), under centain suitable conditions, the authors obtain such results: on the whole plane, the bi-random Dirichlet series and some Dirichlet series almost surely have the same abscissa of convergence (uniform convergence, or absolute convergence), the same order of growth and (p, q)(R)-order, and the necessary and sufficient conditions of the bi-random Dirichlet series almost surely with the same order after rearrangement of the coefficients.

Key words: φ-mixing random sequences, B-valued bi-random Dirichlet series, Order of growth, Rearrangement, (p, q)(R) order