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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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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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