A Weak Convergence of the Linear Random Field Generated by Associated Randomvariables ℤ2

  • Kim, Tae-Sung (Institute of Basic Natural Science, WonKwang University) ;
  • Ko, Mi-Hwa (Institute of Basic Natural Science, WonKwang University) ;
  • Kim, Hyun-Chull (Department of Mathematics Education Daebul University)
  • Published : 2008.11.30


In this paper we show the weak convergence of the linear random(multistochastic process) field generated by identically distributed 2-parameter array of associated random variables. Our result extends the result in Newman and Wright (1982) to the linear 2-parameter processes as well as the result in Kim and Ko (2003) to the 2-parameter case.


  1. Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, New York
  2. Billingsley, P. (1968). Convergence of Probability Measure, Wiley-Interscience, New York
  3. Bulinski, A. V. and Keane, M. S. (1996). Invariance principle for associated random fields, Journal of Mathematical Sciences, 81, 2905-2911
  4. Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications, Annals of Mathematical Statistics, 38, 1466-1474
  5. Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971). Correlation inequalities on some partially ordered sets, Communications in Mathematical Physics, 22, 89-103
  6. Hannan, E. J. (1970). Multiple Time Series, John Wiley & Sons, New York
  7. Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process, In Proceedings of the Cambridge Philosophical Society, 59, 13-20
  8. Kim, T. S. and Ko, M. H. (2003). On a functional central limit theorem for stationary linear processes generated by associated processes, Bulletin of the Korea Mathematical Society, 40, 715-722
  9. Lebowitz, J. L. (1972). Bounds on the correlations and analyticity properties of ferromagnetic ising spin systems, Communications in Mathematical Physics, 28, 313-321
  10. Marinucci, D. and Poghosyan, S. (2001). Asymptotics for linear random fields, Statistics & Probability Letters, 51, 131-141
  11. Newman, C. M. (1980). Normal fluctuations and the FKG inequalities, Communications in Mathematical Physics, 74, 119-128
  12. Newman, C. M. and Wright, A. L. (1981). An invariance principle for certain dependent sequences, The Annals of Probability, 9, 671-675
  13. Newman, C. M. and Wright, A. L. (1982). Associated random variables and Martingale inequalities, Zeitschrift fur Wahrscheinlichkeitstheorie and verwandte Gebiete, 59, 361-371
  14. Phillips, P. C. B. and Solo, V. (1992). Asymptotics for linear processes, Annals of Statistics, 20, 971-1001