A note on the almost sure central limit theorem for the product of some partial sums

Abstract

Abstract 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 "Equation missing" is a standard normal random variable. MSC:60F15, 60F05.