The law of the iterated logarithm and related strong convergence theorems for Banach space valued random variables

1976 ◽  
pp. 224-314 ◽  
Author(s):  
J. Kuelbs
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Random Variables ◽  
Convergence Theorems ◽  
Iterated Logarithm ◽  
The Law

scholarly journals On strong convergence of arrays

1994 ◽  
Vol 50 (2) ◽  
pp. 219-223 ◽  
Author(s):  
Yong-Cheng Qi
Keyword(s):  
Strong Convergence ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Strong Law ◽  
Iterated Logarithm ◽  
The Law ◽  
Independent And Identically Distributed

In this paper we study almost sure convergence for arrays of independent and identically distributed random variables. We obtain a condition under which Marcinkiewicz's strong law holds and get a rate analogous to the law of the iterated logarithm under a condition weaker than Hu and Weber's.

scholarly journals On the law of the iterated logarithm in the infinite variance case

1980 ◽  
Vol 30 (1) ◽  
pp. 5-14 ◽  
Author(s):  
R. A. Maller
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Infinite Variance ◽  
Iterated Logarithm ◽  
The Law ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

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