The Annals of Probability

A strong invariance principle for associated random fields

Raluca M. Balan

Full-text: Open access


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.

Article information

Ann. Probab., Volume 33, Number 2 (2005), 823-840.

First available in Project Euclid: 3 March 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60G60: Random fields
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Strong invariance principle associated random fields blocking technique quantile transform


Balan, Raluca M. A strong invariance principle for associated random fields. Ann. Probab. 33 (2005), no. 2, 823--840. doi:10.1214/009117904000001071.

Export citation


  • Berkes, I. and Morrow, G. J. (1981). Strong invariance principles for mixing random fields. Z. Wahrsch. Verw. Gebiete 57 15--37.
  • Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29--54.
  • Birkel, T. (1988a). The invariance principle for associated sequences. Stochastic Process. Appl. 27 57--71.
  • Birkel, T. (1988b). On the convergence rate in the central limit theorems for associated processes. Ann. Probab. 16 1685--1698.
  • Birkel, T. (1988c). Moment bounds for associated sequences. Ann. Probab. 16 1184--1193.
  • Bulinski, A. V. (1993). Inequalities for the moments of sums of associated multi-indexed variables. Theory Probab. Appl. 38 342--349.
  • Bulinski, A. V. (1995). Rate of convergence in the central limit theorem for fields of associated random variables. Theory Probab. Appl. 40 136--144.
  • Bulinski, A. V. and Keane, M. S. (1996). Invariance principle for associated random fields. J. Math. Sci. 81 2905--2911.
  • Burton, R. M., Dabrowski, A. R. and Dehling, H. (1986). An invariance principle for weakly associated random vectors. Stochastic Process. Appl. 23 301--306.
  • Burton, R. M. and Kim, T. S. (1988). An invariance principle for associated random fields. Pacific J. Math. 132 11--19.
  • Cox, T. J. and Grimmett, G. (1984). Central limit theorems for associated random variables and the percolation model. Ann. Probab. 12 514--528.
  • 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.
  • Dabrowski, A. R. (1985). A functional law of the iterated logarithm for associated sequences. Statist. Probab. Lett. 3 209--212.
  • 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.
  • Esary, J. D., Proscahn, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38 1466--1474.
  • Kim, T. S. (1996). The invariance principle for associated random fields. Rocky Mountain J. Math. 26 1443--1454.
  • Newman, C. M. (1980). Normal fluctuations and the FKG inequalities. Comm. Math. Phys. 74 119--128.
  • Newman, C. M. and Wright, L. A. (1981). An invariance principle for certain dependent sequences. Ann. Probab. 9 671--675.
  • Newman, C. M. and Wright, L. A. (1982). Associated random variables and martingale inequalities. Z. Wahrsch. Verw. Gebiete 59 361--371.
  • 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.
  • Yu, H. (1996). A strong invariance principles for associated random variables. Ann. Probab. 24 2079--2097.