Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE SEQUENTIAL UNIFORM LAW OF LARGE NUMBERS

  • Bae, Jong-Sig (Department of Mathematics and Institute of Basic Science, Sungkyunkwan University) ;
  • Kim, Sung-Yeun (Department of Mathematics and Institute of Basic Science, Sungkyunkwan University)
  • Published : 2006.08.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let $Z_n(s,\;f)=n^{-1}\;{\sum}^{ns}_{i=1}(f(X_i)-Pf)$ be the sequential empirical process based on the independent and identically distributed random variables. We prove that convergence problems of $sup_{(s,\;f)}|Z_n(s,\;f)|$ to zero boil down to those of $sup_f|Z_n(1,\;f)|$. We employ Ottaviani's inequality and the complete convergence to establish, under bracketing entropy with the second moment, the almost sure convergence of $sup_{(s,\;f)}|Z_n(s,\;f)|$ to zero.

Keywords

References

  1. P. Billingsley, Convergence of probability measure, John Wiely & Sons, New York, 1968
  2. J. DeHardt, Generalizations of the Glivenko-Cantelli theorem, Ann. Math. Statist. 42 (1971), 2050-2055 https://doi.org/10.1214/aoms/1177693073
  3. P. Erdos, On a theorem of Hsu and Robbins, Ann. Math. Statist. 20 (1949), 286-291 https://doi.org/10.1214/aoms/1177730037
  4. P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U. S. A. 33 (1947), 25-31
  5. A. F. Karr, Probability, Springer Texts in Statistics, Springer-Verlag, New York, 1993
  6. S. Shreve, Stochastic calculus and finance, Springer Finance, Springer-Verlag, New York, 2004
  7. S. van de Geer, Empirical processes in M-estimation, Cambridge in Statististical and Probabilistic Mathematics (Cambridge University Press, Cambridge, United Kingdom, 2000
  8. A. W. van der Vaart and J. A. Wellner, Weak convergence and empirical processes with applications to statistics, Springer series in Statistics, Springer-Verlag, New York, 1996