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On Strong Versions of the Central Limit Theorem
Author(s) -
Schatte Peter
Publication year - 1988
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19881370117
Subject(s) - mathematics , sequence (biology) , logarithm , zero (linguistics) , limit (mathematics) , combinatorics , central limit theorem , bounded function , function (biology) , limit of a sequence , interval (graph theory) , law of the iterated logarithm , discrete mathematics , distribution (mathematics) , continuous function (set theory) , mathematical analysis , statistics , linguistics , philosophy , genetics , evolutionary biology , biology
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.