Abstract: Consider the convergence of bi-random Dirchlet series ∑_n=l^∞a_n(ω)e^-λ_n(ω)s, wherean a_n(ω) and λ_n(ω) are random variables, we study the limit properties of ∑_i=lⁿa_n(ω) and λ_n(ω) by thestrong law of large numbers and the central limits theorems. Some simple and explict formulae of the absciasssa of convergence σ_c attained under one of following conditions:(i) 0 < lim_n→∞|(∑_i=lⁿEa_i)/n| ≤ lim_n→∞|(∑_i=lⁿEa_i)/n| < ∞; (ii) {a_n} is a sequence of real or complex independent andequally distributed random variables with finite variances D(a_n); (iii) {a_n} is a sequence of independent random with expectation Ea_n=0 and other suitable conditions.

Key words: Weak Dirichlet series, Bi-random Dirichlet series, The strong law of large numbers, Variance