Abstract: Let {S_k, k≥1} be a random sequence satisfying the almost sure central limit theorem,  and {T_k, k≥1} be a random sequence with T_k→0(or 1) a.s. By using the limit theory, we proved that {S_k+T_k, k≥1} and {S_k/T_k, k≥1} satisfied the almost sure central limit theorem, and gave some examples such as linear processes, selfnormalized sums, error variance estimates in linear models and products of partial sums.

Key words: selfnormalized sum, linear processes, product of partial sum, almost sure central limit theorem