Abstract: Let {X_n,n ≥ 1} be a sequence of random variables taking valnes in S={0,1,2,…} with the joint distribution f_n(x₁,…,x_n),(p(0),p(1),…) a distribution on S,m=Σ_k=1^∞kp(k),p_n(x₁,…x_n)=Π_k=1ⁿp(x_k), and ψ_n=(1/n)Σ_k=1ⁿ(X_k-m). In this paper the limit relation between ψ_n(ω) and the log-likelihood ratio L_n(ω)=ln[p_n(X₁,…,X_n)/f_n(X₁,…,X_n)] is investigated and a kind of strong limit theorem represented by inequalities which we call the strong deviation theorem is obtained. In the proof an approach of applying the tool of generating function together with the method of splitting intervals to the investigation of the strong limit theorem is proposed.

Key words: Strong deviation theorem, Strong limit theorem, Likelihood ratio, Log-likeli-hood ratio, Generating function