Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T11:39:31.412Z Has data issue: false hasContentIssue false
  • English
  • Français

ON FUNCTIONAL CENTRAL LIMIT THEOREMS FOR LINEAR RANDOM FIELDS WITH DEPENDENT INNOVATIONS

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  01 April 2008

MI-HWA KO ,
HYUN-CHULL KIM  and
TAE-SUNG KIM
MI-HWA KO
Affiliation:
Department of Mathematics, WonKwang University, Jeonbuk, 570-749, Korea (email: songhack@wonkwang.ac.kr)
HYUN-CHULL KIM
Affiliation:
Department of Mathematics Education, Daebul University, 526-720, Korea (email: kimhc@mail.daebul.ac.kr)
TAE-SUNG KIM*
Affiliation:
Department of Mathematics, WonKwang University, Jeonbuk, 570-749, Korea (email: starkim@wonkwang.ac.kr)
*
✉For correspondence; e-mail: starkim@wonkwang.ac.kr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a linear random field (linear p-parameter stochastic process) generated by a dependent random field with zero mean and finite qth moments (q>2p), we give sufficient conditions that the linear random field converges weakly to a multiparameter standard Brownian motion if the corresponding dependent random field does so.

Keywords

functional central limit theoremrandom fieldassociatedlinear random fieldsmartingale differencedecomposition

MSC classification

Secondary: 60F17: Functional limit theorems; invariance principles 60G15: Gaussian processes
Type
Research Article
Information
The ANZIAM Journal , Volume 49 , Issue 4 , April 2008 , pp. 533 - 541
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Bickel, P. J. and Wichura, M. J., “Convergence criteria for multiparameter stochastic processes and some applications”, Ann. Math. Statist. 42 (1971) 1650–1670.CrossRefGoogle Scholar
[2]Bulinski, A. V., “Invariance principle for associated random fields”, J. Math. Sci. 81 (1996) 2905–2911.CrossRefGoogle Scholar
[3]Burkholder, D. L., “Distribution function inequalities for martingale”, Ann. Probab. 1 (1973) 19–42.CrossRefGoogle Scholar
[4]Burton, R. M. and Kim, T. S., “An invariance principle for associated random fields”, Pacific J. Math. 132 (1988) 11–19.CrossRefGoogle Scholar
[5]Chen, D., “A uniform central limit theorem for nonuniform ϕ-mixing random fields”, Ann. Probab. 19 (1991) 636–649.CrossRefGoogle Scholar
[6]Cox, J. T. and Grimmett, G., “Central limit theorem for associated random variables and the percolation model”, Ann. Probab. 12 (1984) 514–528.CrossRefGoogle Scholar
[7]Goldie, C. M. and Greenwood, P. E., “Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes”, Ann. Probab. 14 (1986) 817–839.CrossRefGoogle Scholar
[8]Marinucci, M. and Poghosyan, S., “Asymptotics for linear random fields”, Statist. Probab. Lett. 51 (2001) 131–141.CrossRefGoogle Scholar
[9]Phillips, P. C. B. and Solo, V., “Asymptotics for linear processes”, Ann. Statist. 20 (1992) 971–1001.CrossRefGoogle Scholar
[10]Poghosyan, S. and Roelly, S., “Invariance principle for martingale-difference random fields”, Statist. Probab. Lett. 38 (1998) 235–245.CrossRefGoogle Scholar
[11]Straf, M. L., “Weak convergence of stochastic processes with several parameters”, in Proc. Sixth Berkeley Symp. on mathematical statistics and probability, Volume 11 (University of California Press, 1970) 187–221.Google Scholar