Law of the iterated logarithm for \(\phi\)-mixing random variables
type: Item
instance of: scholarly article
title: Law of the iterated logarithm for \(\phi\)-mixing random variables
zbMATH Open document ID: 0569.60028
DOI: 10.1007/BF00969519
author: S. A. Utev
published in: Siberian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: Let \(\{\xi_ i\}^{\infty}_{i=1}\) be a sequence of identically distributed random variables with zero expectation and finite variance. We put \(M^ b_ a=\sigma (\xi_ i,\quad a\leq i\leq b),\quad \phi (k)=\sup_{1\leq i<\infty}\sup_{A\in M^ i_ 1, B\in M^{\infty}_{i+k}}| P(B| A)-P(B)|,\quad S_ k=\sum^{k}_{i=1}\xi_ i,\) \(k=1,2,...\), \(\phi (0)=1\), \(\sigma_ n=DS_ n\). The basic result of this note is expressed in the following theorem. Let the following conditions be satisfied: 1) \(\sum^{\infty}_{k=1}\phi^{1/4}(k)k^ 2<\infty\); 2)\underbar{lim} \(\sigma\) \({}_ n/n>0\). Then the law of the iterated logarithm holds: \(\overline{\lim} S_ n/\sqrt{2\sigma_ n\ln \ln \sigma_ n}=1\) a.s.
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 3909432
zbMATH Keywords: \(\phi \) -mixing random variables
zbMATH Keywords: law of the iterated logarithm
Wikidata QID: Q114852777
MaRDI profile type: Publication
cites work: Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments
cites work: The Law of the Iterated Logarithm for Mixing Stochastic Processes
cites work: Almost sure invariance principles for partial sums of weakly dependent random variables
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cites work: Q4085018
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full work available at URL: https://doi.org/10.1007/bf00969519
OpenAlex ID: W1984078995
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description: scientific article; zbMATH DE number 3909432