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Glivenko-cantelli type theorems for distance functions based on the modified empirical distribution function of M. kac and for the empirical process with random sample size in general

Part of the Lecture Notes in Mathematics book series (LNM,volume 296)

Keywords

  • Central Limit Theorem
  • Independent Random Variable
  • Empirical Process
  • Empirical Distribution Function
  • Poisson Random Variable

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References

  1. Allen, J. L. and Beekman, J. A. (1966). A statistical test involving a random variables. Ann. Math. Statist. 37 1305–1309.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Allen, J. L. and Beekman, J. A. (1967). Distribution of a M. Kac Statistic. Ann. Math. Statist. 38 1919–1923.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Anscombe, F. J. (1952). Large-sample theory of sequential estimation. Proc. Cambridge Philos. Soc. 48 600–607.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Barndorff-Nielsen, O. (1964). On the limit distribution of the maximum of a random number of independent random variables. Acta Math. Acad. Sci. Hung. 15, 399–403.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Billingsley, Patrick (1962). Limit theorems for randomly selected partial sums. Ann. Math. Statist. 33 85–92.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Blum, J. R., Hanson, D. L. and Rosenblatt, J. I (1963). On the central limit theorem for the sum of a random number of independent random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1 389–393.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Csörgő, Miklós (1968). On the strong law of large numbers and the central limit theorem for martingales. Trans. Amer. Math. Soc. 131 259–275.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Csorgo, Miklos and Fischler, Roger (1967). Departure from independence; the strong law, standard and random-sum central limit theorems. (Unpublished)

    Google Scholar 

  9. Hájek, J. and Rényi, A. (1955). Generalization of an inequality of Kolmogorov. Acta Math. Acad. Sci. Hung. 6 281–283.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Kac, M. (1949). On deviations between theoretical and empirical distributions. Proc. Nat. Acad. Sci. U.S.A. 35 252–257.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Mogyoródi, J. (1962). A central limit theorem for the sum of a random number of independent random variables. Publications of the Math. Inst. Hung. Acad. Sci., Series A 7 409–424.

    MathSciNet  MATH  Google Scholar 

  12. Pyke, Ronald (1966). The weak convergence of the empirical process with random sample size. (Unpublished).

    Google Scholar 

  13. Rényi, A. (1957). On the asumptotic distribution of the sum of a random number of independent random variables. Acta Math. Acad. Sci. Hung. 8 193–197.

    CrossRef  MATH  Google Scholar 

  14. Rényi, A. (1960). On the central limit theorem for the sum of a random number of independent random variables. Acta Math. Acad. Sci. Hung. 11 97–102.

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1973 Springer-Verlag

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Csörgő, M. (1973). Glivenko-cantelli type theorems for distance functions based on the modified empirical distribution function of M. kac and for the empirical process with random sample size in general. In: Behara, M., Krickeberg, K., Wolfowitz, J. (eds) Probability and Information Theory II. Lecture Notes in Mathematics, vol 296. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0059823

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  • DOI: https://doi.org/10.1007/BFb0059823

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  • Print ISBN: 978-3-540-06211-0

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