Journal of Theoretical Probability

, Volume 5, Issue 2, pp 355–373 | Cite as

On the spectral density and asymptotic normality of weakly dependent random fields

  • Richard C. Bradley


For weakly stationary random fields, conditions on coefficients of “linear dependence” are given which are, respectively, sufficient for the existence of a continuous spectral density, and necessary and sufficient for the existence of a continuous positive spectral density. For strictly stationary random fields, central limit theorems are proved under the corresponding “unrestricted ϱ-mixing” condition and just finite or “barely infinite” second moments. No mixing rate is assumed.

Key Words

Stationary random fields spectral density ϱ-mixing central limit theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bergh, J., and Löfström, J. (1976).Interpolation Spaces, Springer, New York.Google Scholar
  2. 2.
    Bradley, R. C. (1987). The central limit question under ϱ-mixing.Rocky Mountain J. Math. 17, 95–114.Google Scholar
  3. 3.
    Bradley, R. C. (1988). A central limit theorem for stationary ϱ-mixing sequences with infinite variance.Ann. Prob. 16, 313–332.Google Scholar
  4. 4.
    Bradley, R. C., and Bryc, W. (1985). Multilinear forms and measures of dependence between random variables.J. Multivar. Anal. 16, 335–367.Google Scholar
  5. 5.
    Dvoretzky, A. (1972). Asymptotic normality for sums of dependent random variables.Sixth Berkeley Symp. Math. Stat. Prob. 2, 513–535.Google Scholar
  6. 6.
    Goldie, C. M., and Greenwood, P. (1986). Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes.Ann. Prob. 14, 817–839.Google Scholar
  7. 7.
    Goldie, C. M., and Morrow, G. J. (1986). Central limit questions for random fields. In Eberlein, E., and Taqqu, M. S., (eds.).Dependence in Probability and Statistics, Progress in Probability and Statistics, Vol. 11, Birkhäuser, Boston, pp. 275–289.Google Scholar
  8. 8.
    Gorodetskii, V. V. (1984). The central limit theorem and an invariance principle for weakly dependent random fields.Soviet Math. Dokl. 29, 529–532.Google Scholar
  9. 9.
    Ibragimov, I. A. (1975). A note on the central limit theorem for dependent random variables.Theor. Prob. Appl. 20, 135–141.Google Scholar
  10. 10.
    Ibragimov, I. A., and Linnik, Yu. V. (1971).Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen.Google Scholar
  11. 11.
    Ibragimov, I. A., and Rozanov, Y. A. (1978).Gaussian Random Processes, Springer, New York.Google Scholar
  12. 12.
    Peligrad, M. (1986). Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables (a survey). In Eberlein, E., and Taqqu, M. S. (eds.),Dependence in Probability and Statistics, Progress in Probability and Statistics, Vol. 11, Birkhäuser, Boston, pp. 193–223.Google Scholar
  13. 13.
    Rosenblatt, M. (1985).Stationary Sequences and Random Fields, Birkhäuser, Boston.Google Scholar
  14. 14.
    Sarason, D. (1972). An addendum to “Past and Future”.Math. Scand. 30, 62–64.Google Scholar
  15. 15.
    Shao, Q. (1986).An Invariance Principle for Stationary ϱ-Mixing Sequences with Infinite Variance. Report, Department of Mathematics, Hangzhou University, Hangzhou, Peoples Republic of China.Google Scholar
  16. 16.
    Withers, C. S. (1981). Central limit theorems for dependent random variables.Z. Wahrsch. verw. Gebiete 57, 509–534.Google Scholar
  17. 17.
    Zhurbenko, I. G. (1986).The Spectral Analysis of Time Series, North-Holland, Amsterdam.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Richard C. Bradley
    • 1
    • 2
  1. 1.Department of MathematicsIndiana UniversityBloomington
  2. 2.Center for Stochastic Processes, Department of StatisticsUniversity of North CarolinaChapel Hill

Personalised recommendations