Abstract: Let {X_n, n ≥ 1} be a sequence of random variables taking values in S={1,2,… N} with the joint distribution p(x₁,…,x_n). lim_n inf[p(X₁,…,X_n]^1/n and lim_n sup[p(x₁,…,x_n]^1/n are called the AEP-like limits. In virtue of these limits, some strong deviation theorems, namely a class of strong limit theorems represented by inequalities are obtained.

Key words: AEP-like limit, Strong deviation theorem, Strong limit theeorem represented by inequalities, Asymptotic equipartition property