Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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On the Hàjek-Rènyi-Type Inequality for Conditionally Associated Random Variables

  • Choi, Jeong-Yeol (School of Mathematics and Informational Statistics, Wonkwang University) ;
  • Seo, Hye-Young (School of Mathematics and Informational Statistics, Wonkwang University) ;
  • Baek, Jong-Il (School of Mathematics and Informational Statistics, Wonkwang University)
  • Received : 20110700
  • Accepted : 20111000
  • Published : 2011.11.30
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Abstract

Let {${\Omega}$, $\mathcal{F}$, P} be a probability space and {$X_n{\mid}n{\geq}1$} be a sequence of random variables defined on it. A finite sequence of random variables {$X_i{\mid}1{\leq}i{\leq}n$} is a conditional associated given $\mathcal{F}$ if for any coordinate-wise nondecreasing functions f and g defined on $R^n$, $Cov^{\mathcal{F}}$ (f($X_1$, ${\ldots}$, $X_n$), g($X_1$, ${\ldots}$, $X_n$)) ${\geq}$ 0 a.s. whenever the conditional covariance exists. We obtain the H$\grave{a}$jek-R$\grave{e}$nyi-type inequality for conditional associated random variables. In addition, we establish the strong law of large numbers, the three series theorem, integrability of supremum, and a strong growth rate for $\mathcal{F}$-associated random variables.

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