Journal of Theoretical Probability

, Volume 2, Issue 4, pp 461–474 | Cite as

On conditional weak convergence

  • T. J. Sweeting


In this paper we discuss a number of technical issues associated with conditional weak convergence. The main modes of convergence of conditional probability distributions areuniform, probability, andalmost sure convergence in the conditioning variable. General results regarding conditional convergence are obtained, including details of sufficient conditions for each mode of convergence, and characterization theorems for uniform conditional convergence.

Key Words

Conditional limit theorems uniform weak convergence equicontinuity convergence of random measures 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bartlett, M. S. (1938). The characteristic function of a conditional statistic.J. London Math. Soc. 13, 62–67.Google Scholar
  2. 2.
    Basawa, I. V., and Scott, D. J. (1983).Asymptotic Optimal Inference for Nonergodic Models.Lecture Notes in Statistics. No. 17. Springer, Berlin.Google Scholar
  3. 3.
    Billingsley, P. (1968).Convergence of Probability Measures. Wiley, New York.Google Scholar
  4. 4.
    Boos, D. D. (1985). A converse to Scheffé's theorem.Ann. Stat. 13, 423–427.Google Scholar
  5. 5.
    Dubuc, S., and Seneta, E. (1976). The local limit theorem for the Galton-Watson process.Ann. Prob. 4, 490–496.Google Scholar
  6. 6.
    Holst, L. (1981). Some conditional limit theorems in exponential families.Ann. Prob. 9, 818–830.Google Scholar
  7. 7.
    Kallenberg, O. (1976).Random Measures. Academic Press, New York.Google Scholar
  8. 8.
    Le Cam, L. (1958). Un théorème sur la division d'un intervalle par des points pris au hasard.Publ. Inst. Stat. Univ. Paris 7, 7–16.Google Scholar
  9. 9.
    Parzen, E. (1954). On uniform convergence of families of sequences of random variables.Univ. Calif. Publ. Stat. 2, 23–53.Google Scholar
  10. 10.
    Sethuraman, J. (1961). Some limit theorems for joint distributions.Sankhya, A 23, 379–386.Google Scholar
  11. 11.
    Steck, G. (1957). Limit theorems for conditional distributions.Univ. Calif. Publ. Stat. 42, 237–284.Google Scholar
  12. 12.
    Sweeting, T. J. (1980). Uniform asymptotic normality of the maximum likelihood estimator.Ann. Stat. 8, 1375–1381.Google Scholar
  13. 13.
    Sweeting, T. J. (1986a). Asymptotic conditional inference for the offspring mean of a supercritical Galton-Watson process.Ann. Stat. 14, 925–933.Google Scholar
  14. 14.
    Sweeting, T. J. (1986b). On a converse to Scheffé's theorem.Ann. Stat. 14, 1252–1256.Google Scholar
  15. 15.
    Sweeting, T. J. (1988a). Asymptotic ancillarity and conditional inference for stochastic processes. Preprint.Google Scholar
  16. 16.
    Sweeting, T. J. (1988b). Convergence of conditional probability measures. University of Surrey Technical Reports in Statistics, No. 68.Google Scholar
  17. 17.
    Zabell, S. (1979). Continuous versions of regular conditional distributions.Ann. Prob. 7, 159–165.Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • T. J. Sweeting
    • 1
  1. 1.Department of MathematicsUniversity of SurreyGuildfordEngland

Personalised recommendations