On the Hàjek-Rènyi-Type Inequality for Conditionally Associated Random Variables

Title & Authors
On the Hàjek-Rènyi-Type Inequality for Conditionally Associated Random Variables
Choi, Jeong-Yeol; Seo, Hye-Young; Baek, Jong-Il;

Abstract
Let {$\small{{\Omega}}$, $\small{\mathcal{F}}$, P} be a probability space and {$\small{X_n{\mid}n{\geq}1}$} be a sequence of random variables defined on it. A finite sequence of random variables {$\small{X_i{\mid}1{\leq}i{\leq}n}$} is a conditional associated given $\small{\mathcal{F}}$ if for any coordinate-wise nondecreasing functions f and g defined on $\small{R^n}$, $\small{Cov^{\mathcal{F}}}$ (f($\small{X_1}$, $\small{{\ldots}}$, $\small{X_n}$), g($\small{X_1}$, $\small{{\ldots}}$, $\small{X_n}$)) $\small{{\geq}}$ 0 a.s. whenever the conditional covariance exists. We obtain the H$\small{\grave{a}}$jek-R$\small{\grave{e}}$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 $\small{\mathcal{F}}$-associated random variables.
Keywords
Associated random variables;conditional covariance;conditional associated random variables;H$\small{\grave{a}}$jek-R$\small{\grave{e}}$nyi-type inequality;
Language
English
Cited by
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