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. In this paper, under some suitable integrability and smooth-ness conditions on f, we establish the central limit theorems forX k≤N k −1 f(S k /σ√k),where S k is the partial sums of strictly stationary mixing random vari-ables with EX 1 = 0 and σ 2 = EX 21 +2P ∞k=1 EX 1 X 1+k . We also estab-lish an almost sure limit behaviors of the above sums. 1. IntroductionLet {X n ,n≥ 1} be a sequence of random variables on some probabilityspace (Ω,F,P). Let F ba denote the σ-eld generated by the random variablesX a ,X a+1 ,...,X b . For any two σ-eld A,B ⊂ F putρ(A,B) = supˆCov(X,Y)k Xk 2 k Y k 2 : X∈ L 2 (A),Y ∈ L 2 (B)˙,φ(A,B) = sup{| P(B| A) −P(B) |: A∈ A,B∈ B},α(A,B) = sup{| P(AB) −P(A)P(B) |: A∈ A,B∈ B},where and in the sequel k X k p = (E|X| p ) 1/p for 1 ≤ p<∞. The mixingcoecients of the sequence {X n ,n≥ 1} are dened as usualρ(n) = sup k≥1 ρ(F k1 ,F ∞k+n ),φ(n) = sup k≥1 φ(F k1 ,F ∞k+n ),α(n) = sup k≥1 α(F k1 ,F ∞k+n ).The sequence {X n ,n≥ 1} is called ρ-mixing if ρ(n) → 0, φ-mixing or uniformlymixing if φ(n) → 0, α-mixing or strong mixing if α(n) → 0. Well-knownlemmas imply that ρ(n) ≤ 2φ

THE LIMIT THEOREMS UNDER LOGARITHMIC AVERAGES FOR MIXING RANDOM VARIABLES