Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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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
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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.

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References

  1. Ann. Math. Stat. v.42 Convergence criteria for multiparameter stochastic processes and some applications Bickel, P.J.;Wichura, M.J.
  2. Convergence of Probability Measure Billingsley, P.
  3. Stochastic Process. Appl. v.27 The invariance principle for associated processes Birkel, T.
  4. Theory Probab. Appl. v.38 Inequalities for moments of sums of associated multi indexed variables Bulinskii, A.V.
  5. Pacific J. Math. v.132 An invariance principle for associated random fields Burton, R.M.;Kim, T.S.
  6. Ann Prbab. v.15 Scaling limit for associated random measure Burton, R.M.;Waymire, E.
  7. Ann. Probab. v.12 Central limit theorems for associated random variables and the pereolation model Cox, J.T.;Grimmett, G.
  8. Ann. Probab. v.3 A functional central limit theorem for stationary random fields Deo, C.M.
  9. Ann. Math. Stat. v.38 Association of random variables with applicationn Esary, J.;Proschan, F.;Walkup, D.
  10. Comm. Korean Math. Soc. v.8 The invariance principle for two-parameter associated processes Kim, T.S.;Han
  11. Comm. Math. Phys. v.74 Normal fluctuations and the FKG inequalities Newman, C.M.
  12. Ann. Probab. v.9 An invariance principle for certain dependent sequences Newman, C.M.;Wright, A.L.
  13. Asymptotic independence and limit theorems for positively and negatively dependent random variables, IMS Lecture Notes, Monograph Series(ed. Tong), 5 Newman, C.M.