previous :: next
A strong invariance principle for associated random fields
Raluca M. Balan
Source: Ann. Probab. Volume 33, Number 2
(2005), 823-840.
Abstract
In this paper we generalize Yu’s [Ann. Probab. 24 (1996) 2079–2097] strong invariance principle for associated sequences to the multi-parameter case, under the assumption that the covariance coefficient u(n) decays exponentially as n→∞. The main tools that we use are the following: the Berkes and Morrow [Z. Wahrsch. Verw. Gebiete 57 (1981) 15–37] multi-parameter blocking technique, the Csörgő and Révész [Z. Wahrsch. Verw. Gebiete 31 (1975) 255–260] quantile transform method and the Bulinski [Theory Probab. Appl. 40 (1995) 136–144] rate of convergence in the CLT.
First Page:
Show
Hide
Keywords: Strong invariance principle; associated random fields; blocking technique; quantile transform
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1109868602
Digital Object Identifier: doi:10.1214/009117904000001071
Mathematical Reviews number (MathSciNet): MR2123212
Zentralblatt MATH identifier: 02164484
References
Berkes, I. and Morrow, G. J. (1981). Strong invariance principles for mixing random fields. Z. Wahrsch. Verw. Gebiete 57 15--37.
Mathematical Reviews (MathSciNet): MR623453
Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29--54.
Mathematical Reviews (MathSciNet): MR515811
JSTOR: links.jstor.org
Birkel, T. (1988a). The invariance principle for associated sequences. Stochastic Process. Appl. 27 57--71.
Mathematical Reviews (MathSciNet): MR934529
Digital Object Identifier: doi:10.1016/0304-4149(87)90005-6
Zentralblatt MATH: 0632.60001
Birkel, T. (1988b). On the convergence rate in the central limit theorems for associated processes. Ann. Probab. 16 1685--1698.
Mathematical Reviews (MathSciNet): MR958210
JSTOR: links.jstor.org
Birkel, T. (1988c). Moment bounds for associated sequences. Ann. Probab. 16 1184--1193.
Mathematical Reviews (MathSciNet): MR942762
JSTOR: links.jstor.org
Bulinski, A. V. (1993). Inequalities for the moments of sums of associated multi-indexed variables. Theory Probab. Appl. 38 342--349.
Mathematical Reviews (MathSciNet): MR1317986
Bulinski, A. V. (1995). Rate of convergence in the central limit theorem for fields of associated random variables. Theory Probab. Appl. 40 136--144.
Mathematical Reviews (MathSciNet): MR1346738
Bulinski, A. V. and Keane, M. S. (1996). Invariance principle for associated random fields. J. Math. Sci. 81 2905--2911.
Mathematical Reviews (MathSciNet): MR1420893
Digital Object Identifier: doi:10.1007/BF02362501
Zentralblatt MATH: 0898.60043
Burton, R. M., Dabrowski, A. R. and Dehling, H. (1986). An invariance principle for weakly associated random vectors. Stochastic Process. Appl. 23 301--306.
Mathematical Reviews (MathSciNet): MR876052
Digital Object Identifier: doi:10.1016/0304-4149(86)90043-8
Zentralblatt MATH: 0611.60028
Burton, R. M. and Kim, T. S. (1988). An invariance principle for associated random fields. Pacific J. Math. 132 11--19.
Mathematical Reviews (MathSciNet): MR929580
Cox, T. J. and Grimmett, G. (1984). Central limit theorems for associated random variables and the percolation model. Ann. Probab. 12 514--528.
Mathematical Reviews (MathSciNet): MR735851
JSTOR: links.jstor.org
Csörgő, M. and Révész, P. (1975). A new method to prove Strassen type laws of invariance principle I. Z. Wahrsch. Verw. Gebiete 31 255--260.
Mathematical Reviews (MathSciNet): MR375411
Dabrowski, A. R. (1985). A functional law of the iterated logarithm for associated sequences. Statist. Probab. Lett. 3 209--212.
Mathematical Reviews (MathSciNet): MR801691
Dabrowski, A. R. and Dehling, H. (1988). A Berry--Essen theorem and a functional law of the iterated logarithm for weakly associated random variables. Stochastic Process. Appl. 30 277--289.
Mathematical Reviews (MathSciNet): MR978359
Digital Object Identifier: doi:10.1016/0304-4149(88)90089-0
Zentralblatt MATH: 0665.60027
Esary, J. D., Proscahn, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38 1466--1474.
Mathematical Reviews (MathSciNet): MR217826
Digital Object Identifier: doi:10.1214/aoms/1177698701
Kim, T. S. (1996). The invariance principle for associated random fields. Rocky Mountain J. Math. 26 1443--1454.
Mathematical Reviews (MathSciNet): MR1447596
Newman, C. M. (1980). Normal fluctuations and the FKG inequalities. Comm. Math. Phys. 74 119--128.
Mathematical Reviews (MathSciNet): MR576267
Digital Object Identifier: doi:10.1007/BF01197754
Zentralblatt MATH: 0429.60096
Newman, C. M. and Wright, L. A. (1981). An invariance principle for certain dependent sequences. Ann. Probab. 9 671--675.
Mathematical Reviews (MathSciNet): MR624694
JSTOR: links.jstor.org
Newman, C. M. and Wright, L. A. (1982). Associated random variables and martingale inequalities. Z. Wahrsch. Verw. Gebiete 59 361--371.
Mathematical Reviews (MathSciNet): MR721632
Philipp, W. and Stout, W. F. (1975). Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 161.
Wichura, M. J. (1973). Some Strassen-type laws of the iterated logarithm for multiparameter stochastic processes with independent increments. Ann. Probab. 1 272--296.
Mathematical Reviews (MathSciNet): MR394894
Digital Object Identifier: doi:10.1214/aop/1176996980
Yu, H. (1996). A strong invariance principles for associated random variables. Ann. Probab. 24 2079--2097.
Mathematical Reviews (MathSciNet): MR1415242
Digital Object Identifier: doi:10.1214/aop/1041903219
Project Euclid: euclid.aop/1041903219
Zentralblatt MATH: 0879.60028
previous :: next
The Annals of Probability
