# A FUNCTIONAL CENTRAL LIMIT THEOREM FOR ASSOCIATED RANDOM FIELD

• KIM, TAE-SUNG (Division of Mathematics & Statistics and Institute of Basic Natural Science, WonKwang University) ;
• KO, MI-HWA (Division of Mathematics & Statistics and Institute of Basic Natural Science, WonKwang University)
• Received : 2002.03.28
• Published : 2002.07.30
• 38 5

#### Abstract

In this paper we prove a functional central limit theorem for a field $\{X_{\underline{j}}:{\underline{j}}{\in}Z_+^d\}$ of nonstationary associated random variables with $EX{\underline{j}}=0,\;E{\mid}X_{\underline{j}}{\mid}^{r+{\delta}}<{\infty}$ for some $r>2,\;{\delta}>0$and $u(n)=O(n^{-{\nu}})$ for some ${\nu}>0$, where $u(n):=sup_{{\underline{i}}{\in}Z_+^d{\underline{j}}:{\mid}{\underline{j}}-{\underline{i}}{\mid}{\geq}n}{\sum}cov(X_{\underline{i}},\;X_{\underline{j}}),\;{\mid}{\underline{x}}{\mid}=max({\mid}x_1{\mid},{\cdots},{\mid}x_d{\mid})\;for\;{\underline{x}}{\in}{\mathbb{R}}^d$. Our investigation implies and analogous result in the case associated random measure.

#### Keywords

associated;random field;functional central limit theorem;random measure

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