Abstract: Let {ε_t,t∈Z} be a strictly stationary sequence defined on a probability space (Ω,F,P) such that Eε₀= 0, and E|ε₀|^p<∞, for some p >2. And the sequence {ε_t,t∈Z}is assumed to satisfy the strong mixingconditions. {a_j, j∈Z}is a sequence of real numbers with . Using the weak convergence theorem of the linear process generated by strong mixing sequences and the moment inequalities of strong mixing sequences, we studied the precise asymptotics of a kind of weighted infinite series ofunder the conditions of b_n=O(1/log log n).

Key words: strong mixing sequences, linear process, the law of the iterated logarithm, precise asym ptotics