Probability Theory and Related Fields

, Volume 88, Issue 3, pp 381–404 | Cite as

Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales

  • Ditlev Monrad
  • Walter Philipp
Article

Summary

In this paper we focus on sequences of random vectors which do not admit a strong approximation of their partial sums by sums of independent random vectors. In the first part we prove conditional versions of the Strassen-Dudley theorem. We apply these in the second part of the paper to obtain strong invariance principles for vector-valued martingales which, when properly normalized, converge in law to a mixture of Gaussian distributions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berger, E.: personal communication (1982)Google Scholar
  2. 2.
    Berkes, I., Philipp, W.: Approximation for independent and weakly dependent random vectors. Ann. Probab.7, 29–54 (1979)Google Scholar
  3. 3.
    Billingsley, P.: Probability and measure, 2nd edn., New York: Wiley 1986Google Scholar
  4. 4.
    Dudley, R.M.: Real Analysis and Probability, Wadsworth, Belmont, California (1989)Google Scholar
  5. 5.
    Dudley, R.M., Philipp, W.: Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.62, 509–552 (1983)Google Scholar
  6. 6.
    Eberlein, E.: On strong invariance principles under dependence assumptions. Ann. Probab.14, 260–270 (1986)Google Scholar
  7. 7.
    Eberlein, E.: Strong approximation of continuous time stochastic processes. J. Multivariate Anal.31, 220–235 (1989)Google Scholar
  8. 8.
    Monrad, D., Philipp, W.: The problem of embedding vector-valued martingales in a Gaussian process. Teor. Veroyatn. Primen.35, 384–387 (1990)Google Scholar
  9. 9.
    Morrow, G.J., Philipp, W.: An almost sure invariance principle for Hilbert space valued martingales. Trans. Am. Math. Soc.273, 231–251 (1982)Google Scholar
  10. 10.
    Philipp, W.: Invariance principles for independent and weakly dependent random variables. In: Eberlein, E., Taqqu, M.S. (eds.) Dependence in probability and statistics, pp. 225–268. Boston: Birkhäuser 1986Google Scholar
  11. 11.
    Philipp, W.: A note on the almost sure approximation of weakly dependent random variables. Monatsh. Math.102, 227–236 (1986)Google Scholar
  12. 12.
    Strassen, V.: Almost sure behavior of sums of independent random variables and martingales. Proc. Fifth Berkeley Symp. Math. Stat. Prob. Vol. II, Part 1, pp. 315–343 (1967)Google Scholar
  13. 13.
    Yosida, K.: Functional analysis. (Grundlehren der mathematischen Wissenschaften, Bd. 123) 3rd edn., Berlin Heidelberg New York: Springer 1971Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Ditlev Monrad
    • 1
  • Walter Philipp
    • 1
  1. 1.Departments of Mathematics and StatisticsUniversity of IllinoisUrbanaUSA

Personalised recommendations