On Strong Versions of the Central Limit Theorem

Abstract

Abstract Let S n be the sum of n i.i.d.r.v. and let 1 (‐∞, x ) (·) be the indicator function of the interval (‐∞, x ). Then the sequence 1 (‐∞, x ) ( S n /√n) does not converge for any x. Likewise the arithmetic means of this sequence converge only with probability zero. But the logarithmic means converge with probability one to the standard normal distribution Ø( x ). Then for any bounded and a.e. continuous function a ( y ) the logarithmic means of a ( S n /√n) converge a.s. to a = ∫ a ( y ) d Ø( y ). The arithmetic means of a ( S nk /√n) converge to the same limit a for all subsequences n k = [ c k ], c > 1.