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Convergence of stochastic empirical measures

Author

Listed:
  • Beran, R.J.
  • Le Cam, L.
  • Millar, P.W.

Abstract

Let Pn be a random probability measure on a metric space S. Let P^n be the empirical measure of kn iid random variables, each distributed according to Pn. Our main theorem asserts that if {Pn} converges in distribution, as random probability measures on S, then so does {P^n}. Applications of the result to the study of bootstrap and other stochastic procedures are given.

Suggested Citation

  • Beran, R.J. & Le Cam, L. & Millar, P.W., 1987. "Convergence of stochastic empirical measures," Journal of Multivariate Analysis, Elsevier, vol. 23(1), pages 159-168, October.
    • Handle: RePEc:eee:jmvana:v:23:y:1987:i:1:p:159-168
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    Citations

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    Cited by:

    1. Natalie Neumeyer & Ingrid Van Keilegom, 2009. "Change‐Point Tests for the Error Distribution in Non‐parametric Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(3), pages 518-541, September.
    2. Natalie Neumeyer, 2009. "Smooth Residual Bootstrap for Empirical Processes of Non‐parametric Regression Residuals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 204-228, June.
    3. Shao, Yongzhao, 2001. "Rate of convergence of bootstrapped empirical measures," Statistics & Probability Letters, Elsevier, vol. 53(3), pages 293-298, June.
    4. Delgado, Miguel A. & Fiteni, Inmaculada, 2002. "External bootstrap tests for parameter stability," Journal of Econometrics, Elsevier, vol. 109(2), pages 275-303, August.
    5. Delicado, Pedro, 1995. "Random coefficient regressions: parametric goodness of fit tests," DES - Working Papers. Statistics and Econometrics. WS 4199, Universidad Carlos III de Madrid. Departamento de Estadística.
    6. Axel Bücher & Ivan Kojadinovic, 2019. "A Note on Conditional Versus Joint Unconditional Weak Convergence in Bootstrap Consistency Results," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1145-1165, September.

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