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Almost sure central limit theorem for partial sums and maxima
Author(s) -
Zuoxiang Peng,
Lili Wang,
Nadarajah Saralees
Publication year - 2009
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.200610760
Subject(s) - mathematics , lipschitz continuity , maxima , bounded function , limit (mathematics) , combinatorics , distribution (mathematics) , central limit theorem , function (biology) , random variable , image (mathematics) , mathematical analysis , statistics , art , evolutionary biology , performance art , biology , art history , artificial intelligence , computer science
Let X , X 1 , X 2 , … be i.i.d. random variables with nondegenerate common distribution function F , satisfying EX = 0, EX 2 = 1. Let X i and M n = max{ X i , 1 ≤ i ≤ n }. Suppose there exists constants a n > 0, b n ∈ R and a nondegenrate distribution G ( y ) such thatThen, we havealmost surely, where f ( x , y ) denotes the bounded Lipschitz 1 function and Φ( x ) is the standard normal distribution function (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)