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A Nonclassical Law of the Iterated Logarithm for Functions of Positively Associated Random Variables

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Abstract

Let \(\{X_{n}; n\geq 1\}\) be a sequence of stationary positively associated random variables and a sequence of positive constants \(\{b(n); n\geq1\}\) be monotonically approaching infinity and be not asymptotically equivalent to loglog n. Under some suitable conditions, a nonclassical law of the iterated logarithm is investigated, i.e.

$$\limsup_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}[f(X_{i})-E f(X_{i})]}{\sqrt{2nb(n)}}=\sigma_{f}\hspace{0.3cm} a.s., $$

where (f) is a real function and \(\sigma_{f}^{2}=Var(f(X_{1}))+2\sum_{j=2}^{\infty}Cov(f(X_{1}), f(X_{j}))\).

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References

  • Balan RM (2005) A strong invariance principle for associated random fields. Ann Probab 33:823–840

    Article  MATH  MathSciNet  Google Scholar 

  • Birkel T (1988a) Moment bounds for associated sequences. Ann Probab 16:1184–1193

    MathSciNet  Google Scholar 

  • Birkel T (1988b) On the convergence rate in the central limit theorem for associated processes. Ann Probab 16:1685–1698

    MathSciNet  Google Scholar 

  • Burton RM, Dabrowski AR, Dehling H (1986) An invariance principle for weakly associated random vectors. Stochastic Process Appl 23:301–306

    Article  MATH  MathSciNet  Google Scholar 

  • Cai Z, Roussas GG (1999a). Weak convergence for a smooth estimator of a distribution function under association. Stochastic Anal Appl 17:145–168

    MathSciNet  Google Scholar 

  • Cai Z, Roussas GG (1999b) Berry–Esseen bounds for smooth estimator of a distribution function under association. J Nonparametric Statist 10:79–106

    MathSciNet  Google Scholar 

  • Dabrowski AR, Dehling H (1988) A Berry–Esseen theorem and a functional law of the iterated logrithm for weakly associated random vectors. Stochastic Process Appl 30:277–289

    Article  MATH  MathSciNet  Google Scholar 

  • Esary J, Proschan F, Walkup D (1967) Association of random variables with applications. Ann Math Statist 38:1466–1474

    MATH  MathSciNet  Google Scholar 

  • Huang W (2004) A nonclassical law of the iterated logarithm for functions of negatively associated random variables. Stochastic Anal Appl 22:657–678

    Article  MATH  MathSciNet  Google Scholar 

  • Kesten H (1972) Sums of independent random variables-without moment conditions. Ann Math Statist 43:701–732

    MATH  MathSciNet  Google Scholar 

  • Klesov O, Rosalsky A (2001) A nonclassical law of the iterated logarithm for i.i.d. square integrable random variables. Stochastic Anal Appl 19:627–641

    Article  MATH  MathSciNet  Google Scholar 

  • Lin ZY (1983) On generalizations of Berry–Esseen inequality for U-statistics. Acta Math Applica Sinica (in Chinese) 6:468–475

    MATH  Google Scholar 

  • Newman CM, Wright AL (1981) An invariance principle for certain dependent sequences. Ann Probab 9:671–675

    MATH  MathSciNet  Google Scholar 

  • Peligrad M, Shao QM (1995) Estimation of the variance of partial sums for (ρ)-mixing random variables. J Multivariate Anal 52:140–157

    Article  MATH  MathSciNet  Google Scholar 

  • Peligrad M, Suresh R (1995) Estimation of variance of partial sums of an associated sequence of random variables. Stochastic Process Appl 56:307–319

    Article  MATH  MathSciNet  Google Scholar 

  • Petrov VV (1975) Sums of independent random variables. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  • Prakasa Rao BLS (2002) Hajeck-Renyi-type inequality for associated sequences. Statist Probab Lett 57:139–143

    Article  MATH  MathSciNet  Google Scholar 

  • Rosalsky A (1981) A generalization of the iterated logarithm law for weighted sums with infinite variance. Z Wahrsch Verw Gebiete 58:351–372

    Article  MATH  MathSciNet  Google Scholar 

  • Wang Q, Lin YX, Gulati CM (2003) Strong approximation for long memory processes with applications. J Theor Probab 16:377–389

    Article  MATH  MathSciNet  Google Scholar 

  • Wood TE (1983) A Berry–Esseen theorem for associated random variables. Ann Probab 11:1042–1047

    MATH  MathSciNet  Google Scholar 

  • Yu H (1993) A Glivenko centelli lemma and weak convergence for empirical processes of associated sequences. Probab Theory Related Fields 95:357–370

    Article  MATH  MathSciNet  Google Scholar 

  • Yu H (1996) An strong invariance principle for associated sequences. Ann Probab 24:2079–2097

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang LX (2001) The weak convergence for functions of negatively associated random variables. J Multivariate Anal 78:272–298

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics and Computing Science, Zhejiang Gongshang University, Hangzhou, 310035, P.R. China

    Jian-Feng Wang

  2. Department of Mathematics, Zhejiang Unviersity, Xixi Campus, Hangzhou, 310028, P.R. China

    Li-Xin Zhang

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  1. Jian-Feng Wang
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  2. Li-Xin Zhang
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Correspondence to Jian-Feng Wang.

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Wang, JF., Zhang, LX. A Nonclassical Law of the Iterated Logarithm for Functions of Positively Associated Random Variables. Metrika 64, 361–378 (2006). https://doi.org/10.1007/s00184-006-0054-y

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  • DOI: https://doi.org/10.1007/s00184-006-0054-y

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