Abstract: Let {X_n, n≥1} be an independent and identically distributed continuous random variables， let {L_n, n≥1} and {μ_n, n≥1} be the record times sequence and corresponding counting process. By using the central limit theorem, the moment inequalities and the Berry-Esseen inequalities of the record times and its counting process, a general result of precise asymptotics for the record times and corresponding counting process were obtained for the boundary function and quasi-weight function.

Key words: counting process, record times, precise asymptotics, general form