Skip to main content
Log in

Some Laws of the Iterated Logarithm for Two Parameter Martingales

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

We prove some laws of the iterated logarithm for two parameter martingale differences of the following type: E(X t|σ(X s, s 1<t 1 or s 2<t 2))=0 a.s.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. Alexander, K. S., and Pyke, R. (1986). A uniform central limit theorem for setindexed partial-sum processes with finite variance. Ann. Probab. 14, 582–597.

    Google Scholar 

  2. Bass, R. F., and Pyke, R. (1984). Functional law of the iterated logarithm and uniform central limit theorem for partial-sum processes indexed by sets. Ann. Probab. 12, 13–34.

    Google Scholar 

  3. Cairoli, R. and Walsh, J. B. (1975). Stochastic integrals in the plane. Acta Math. 134, 111–183.

    Google Scholar 

  4. Hall, P. G., and Heyde, C. C. (1980). Martingale Limit Theory and Its Application, Academic Press, New York.

    Google Scholar 

  5. Helson, H., and Lowdenslarger, D. (1958). Prediction theory and Fourier series in several variables. Acta Math. 19, 165–202.

    Google Scholar 

  6. Heyde, C. C. (1988). Some thoughts on stationary processes and linear time series analysis. A celebration of applied probability. J. Appl. Probab. 25, 309–318.

    Google Scholar 

  7. Huang, D. (1992a). Uniform convergence rates for two-parameter martingales with application to stationary random fields, Peking University. Acta Scientiarum Naturalium 28, 291–302.

    Google Scholar 

  8. Huang, D. (1992b). Central limit theorem for two-parameter martingale differences with application to stationary random fields. Science in China (Ser. A) 35, 413–425.

    Google Scholar 

  9. Huang, D., and Anh, V. V. (1992). Estimation of spatial ARMA models. Austral. J. Statist. 34, 513–530.

    Google Scholar 

  10. Imkeller, P. (1988). Two-parameter martingales and their quadratic variation. Lecture Notes in Math., Vol. 1308, Springer-Verlag.

  11. Jiang, J. (1991a). Uniform convergence rate of sample ACV and ACR for linear spatial series under more general martingale condition. Adv. Math. (China) 20, 39–50.

    Google Scholar 

  12. Jiang, J. (1991b). Parameter estimation of spatial AR model. Chinese Ann. Math. 12B, 432–444.

    Google Scholar 

  13. Jiang, J. (1993). Estimation of spatial AR models. Acta Math. Appl, Sinica 9, 174–187.

    Google Scholar 

  14. Li, D., and Wu, Z. (1989). The law of the iterated logarithm for B-valued random variables with multidimensional indices. Ann. Probab. 17, 760–774.

    Google Scholar 

  15. Móricz, F. (1980). Exponential estimates for the maximum of partial sums II (random fields). Acta Math. Acad. Sci. Hungar. 35, 361–377.

    Google Scholar 

  16. Smythe, R. T. (1973). Strong laws of large numbers for r-dimensional arrays of random variables. Ann. Probab. 1, 164–170.

    Google Scholar 

  17. Soltani, A. R. (1984). Extrapolation and moving average representation for stationary random fields and Beurling's theorem. Ann. Probab. 12, 120–132.

    Google Scholar 

  18. Stout, W. F. (1970). A martingale analogue of Kolmogorov's law of the iterated logarithm. Z. Wahrsch. Verw. Geb. 15, 279–290.

    Google Scholar 

  19. Stout, W. F. (1974). Almost Sure Convergence, Academic Press, New York.

    Google Scholar 

  20. Tjøstheim, D. (1984). Statistical spatial series modelling II: Some further results on unilateral lattice processes. Adv. Appl. Probab. 15, 562–684.

    Google Scholar 

  21. Wichura, M. (1973). Some Strassen-type laws of the iterated logarithm for multiparameter stochastic process with independent increments. Ann. Probab. 1, 272–291.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jiang, J. Some Laws of the Iterated Logarithm for Two Parameter Martingales. Journal of Theoretical Probability 12, 49–74 (1999). https://doi.org/10.1023/A:1021740425864

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021740425864

Navigation