Let ( X n ) be a sequence of i.i.d., positive, square integrable random variables with E ( X 1 ) = μ > 0 , Var ( X 1 ) = σ 2 . Denote by S n , k = ∑ i = 1 n X i ? X k and by γ = σ / μ the coefficient of variation. Our goal is to show the unbounded, measurable functions g, which satisfy the almost sure central limit theorem, i.e., lim N → ∞ 1 log N ∑ n = 1 N 1 n g ( ( ∏ k = 1 n S n , k ( n ? 1 ) n μ n ) 1 γ n ) = ∫ 0 ∞ g ( x ) d F ( x ) a.s. , where F ( ? ) is the distribution function of the random variable e N and is a standard normal random variable. MSC:60F15, 60F05.
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机译：设（X n）是一个序列，即E（X 1）=μ> 0，Var（X 1）=σ2的正，平方可积随机变量。用S n表示，k = ∑ i = 1 n X i？ X k和由γ=σ/μ构成的变异系数。我们的目标是证明满足几乎确定的中心极限定理的无界可测量函数g，即lim N→∞1 log N ∑ n = 1 N 1 ng（（∏ k = 1 n S n，k（n ？1）nμn）1γn）=∫0∞g（x）d F（x）为，其中F（？）是随机变量e N的分布函数，并且是标准正态随机变量。 MSC：60F15、60F05。
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