Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



An inequality for the Lévy distance between two distribution functions and its applications

Authors: K.-H. Indlekofer, O. I. Klesov and J. G. Steinebach
Translated by: The authors
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 81 (2010).
Journal: Theor. Probability and Math. Statist. 81 (2010), 59-70
MSC (2010): Primary 60J05
Published electronically: January 18, 2011
MathSciNet review: 2667310
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a nonuniform bound for the deviation between two distribution functions expressed in terms of the Lévy distance. Applications of this bound to the global version of the central limit theorem are given and complete convergence is shown.

References [Enhancements On Off] (What's this?)

  • 1. Alejandro de Acosta and Evarist Giné, Convergence of moments and related functionals in the general central limit theorem in Banach spaces, Z. Wahrsch. Verw. Gebiete 48 (1979), no. 2, 213–231. MR 534846, 10.1007/BF01886874
  • 2. Ralph Palmer Agnew, Global versions of the central limit theorem, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 800–804. MR 0064342
  • 3. Ralph Palmer Agnew, Estimates for global central limit theorems, Ann. Math. Statist. 28 (1957), 26–42. MR 0084227
  • 4. Ralph Palmer Agnew, Asymptotic expansions in global central limit theorems, Ann. Math. Statist. 30 (1959), 721–737. MR 0107919
  • 5. Carl-Gustav Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math. 77 (1945), 1–125. MR 0014626
  • 6. Carl-Gustav Esseen, On mean central limit theorems, Kungl. Tekn. Högsk. Handl. Stockholm, no. 121 (1958), 31. MR 0097111
  • 7. B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. MR 0062975
  • 8. C. C. Heyde, A supplement to the strong law of large numbers, J. Appl. Probability 12 (1975), 173–175. MR 0368116
  • 9. P. L. Hsu and Herbert Robbins, Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U. S. A. 33 (1947), 25–31. MR 0019852
  • 10. K.-H. Indlekofer and O. I. Klesov, A generalisation of a Kolodyazhnyĭ theorem for the Lévy distance, Int. J. Pure Appl. Math. 47 (2008), no. 2, 235–241. MR 2457827
  • 11. A. F. Kolodyazhnyĭ, A generalization of a theorem of Esseen, Vestnik. Leningrad. Univ. 13 (1968), 28-33. (Russian)
  • 12. V. M. Kruglov, Convergence of numeric characteristics of sums of independent random variables and global theorems, Proceedings of the Second Japan-USSR Symposium on Probability Theory (Kyoto, 1972) Springer, Berlin, 1973, pp. 255–286. Lecture Notes in Math., Vol. 330. MR 0445582
  • 13. G. Laube, Weak convergence and convergence in the mean of distribution functions, Metrika 20 (1973), 103–105. MR 0407939
  • 14. Shōichi Nishimura, An inequality for a metric in a random collision process, J. Appl. Probability 12 (1975), 239–247. MR 0381040
  • 15. V. V. \cyr{P}etrov, Summy nezavisimykh sluchainykh velichin, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0322927
  • 16. Valentin V. Petrov, Limit theorems of probability theory, Oxford Studies in Probability, vol. 4, The Clarendon Press, Oxford University Press, New York, 1995. Sequences of independent random variables; Oxford Science Publications. MR 1353441
  • 17. Andrew Rosalsky, A generalization of the global limit theorems of R. P. Agnew, Internat. J. Math. Math. Sci. 11 (1988), no. 2, 365–374. MR 939092, 10.1155/S0161171288000432
  • 18. Ju. P. Studnev and Ju. I. Ignat, A refinement of the central limit theorem and of its global version, Teor. Verojatnost. i Primenen 12 (1967), 562–567 (Russian, with English summary). MR 0215348

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 60J05

Retrieve articles in all journals with MSC (2010): 60J05

Additional Information

K.-H. Indlekofer
Affiliation: Fakultät für Elektrotechnik, Informatik und Mathematik, Institut für Mathematik, Universität Paderborn, Warburger Straße 100, Paderborn 33098, Germany

O. I. Klesov
Affiliation: Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine (KPI), Prospekt Peremogy 37, Kyiv 03056, Ukraine

J. G. Steinebach
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86–90, Köln D–50931, Germany

Keywords: Lévy distance, global version of the central limit theorem, complete convergence
Received by editor(s): September 11, 2009
Published electronically: January 18, 2011
Additional Notes: Supported by a DFG grant
Article copyright: © Copyright 2010 American Mathematical Society