Acta Mathematica Hungarica

, Volume 156, Issue 1, pp 47–55 | Cite as

On the Komlós–Révész SLLN for dependent variables

  • Z. S. Szewczak


The Komlós–Révész strong law of large numbers (SLLN) is extended and proved for two dependent families of random variables.

Key words and phrases

strong law Komlós–Révész law φ ϱ negative dependence Lévy’s equivalence theorem continued fraction 

Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ash R.B.: Probability and Measure Theory. Academic Press, New York (2000)zbMATHGoogle Scholar
  2. 2.
    Billingsley P.: Probability and Measure. 3rd ed. Wiley, New York (1995)zbMATHGoogle Scholar
  3. 3.
    R. C. Bradley, Introduction to Strong Mixing Conditions, vols. I-III, Kendrick Press (Heber City, 2007).Google Scholar
  4. 4.
    Chandra T.K.: Laws of Large Numbers. Narosa Publishing House, New Delhi (2012)Google Scholar
  5. 5.
    Chow Y.S.: Local convergence of martingales and the law of large numbers. Ann. Math. Stat. 36, 552–558 (1965)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Y. S. Chow and H. Teicher, Probability Theory. Independence, Interchangebility, Martingales 3rd ed., Springer Texts in Statistics, Springer-Verlag (New York, 2003).Google Scholar
  7. 7.
    Chung K.-L.: Note on some strong laws of large numbers. Amer. J. Math. 69, 189–192 (1947)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chung K.-L.: A course in Probability Theory. 3rd ed. Academic Press, New York (2001)Google Scholar
  9. 9.
    Cohen G.: On the Komlós–Révész estimation problem for random variables without variances. Acta Sci. Math. (Szeged) 74, 915–925 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cohen G., Lin M.: Almost sure convergence of weighted sums of independent random variables. Contemp. Math. 485, 13–43 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cohn H.: On a class of dependent variables. Rev. Roumaine Math. Pures Appl. 10, 1593–1606 (1965)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dudley R.M.: Real Analysis and Probability. Chapman & Hall, New York (1989)zbMATHGoogle Scholar
  13. 13.
    Hall P., Heyde C.C.: Martingale Limit Theory and its Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  14. 14.
    Iosifescu M., Teodorescu R.: Random Processes and Learning. Springer-Verlag, Berlin–Heidelberg–New York (1969)CrossRefGoogle Scholar
  15. 15.
    Iosifescu M., Kraaikamp S.: Metrical Theory of Continued Fractions. Kluwer Academic Publishers, Dordrecht (2002)CrossRefGoogle Scholar
  16. 16.
    Jamison B., Orey S., Pruitt W.: Convergence of weighted averages of indenpendent random variables. Z. Wahrsch. verw. Gebiete 4, 40–44 (1965)CrossRefGoogle Scholar
  17. 17.
    Knopp K.: Infinite Sequences and Series. Dover Publication Inc, New York (1958)Google Scholar
  18. 18.
    A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag (Berlin, 1933); reprinted: Foundations of the Theory of Probability, 2nd ed., Chelsea Publ. Comp. (New York, 1956); 2nd Russian edition: Fundamental concepts of probability theory, Nauka (Moscow, 1974).Google Scholar
  19. 19.
    Komlós J., Révész P.: On the weighted averages of independent random variables. Magyar Tud. Akad. Mat. Kutató Int. Közl. 9, 583–587 (1965)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kwapień S., Woyczyński W.: Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992)CrossRefGoogle Scholar
  21. 21.
    Lévy P.: Théorie de l’addition des variables aléatoires. Gauthier-Villars, Paris (1937)zbMATHGoogle Scholar
  22. 22.
    Lin Z., Lu C.: Limit Theory for Mixing Dependent Random Variables. Kluwer Academic Press, Boston (1996)zbMATHGoogle Scholar
  23. 23.
    Lin Z.Y., Bai Z.D.: Probability Inequalities. Science Press (Beijing) and Springer, Berlin (2010)zbMATHGoogle Scholar
  24. 24.
    Loève M.: Probability Theory. 2nd ed. Van Nostrand, New York (1960)zbMATHGoogle Scholar
  25. 25.
    Marcinkiewicz J., Zygmund A.: Sur les fonctions indépendantes. Fund. Math. 29, 60–90 (1937)zbMATHGoogle Scholar
  26. 26.
    Matuła P.: A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Probab. Lett. 15, 209–213 (1992)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Petrov V.V.: Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford Studies in Probability 4, Oxford (1995)zbMATHGoogle Scholar
  28. 28.
    Révész P.: The Laws of Large Numbers. Academic Press, New York (1968)zbMATHGoogle Scholar
  29. 29.
    Rosalsky A.: A strong law for weighted averages of random variables and the Komlós–Révész estimation problem. Calcutta Statist. Assoc. Bull. 35, 59–66 (1986)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Shao Q.-M.: A comparison theorem of on moment inequalities between negatively associated and andependent random variables. J. Theoret. Probab. 13, 343–356 (2000)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Shiryaev A.N.: Probability. 2nd ed. Springer, New York (1996)CrossRefGoogle Scholar
  32. 32.
    Stout W.F.: On convergence of φ-mixing sequences of random variables. Z. Wahrsch. verw. Gebiete 31, 69–70 (1974)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Stout W.F.: Almost Sure Convergence. Academic Press, New York (2000)zbMATHGoogle Scholar
  34. 34.
    Szewczak Z.S.: limit theorems for continued fractions. J. Theoret. Probab. 22, 239–255 (2009)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Szewczak Z.S.: Marcinkiewicz laws with infinite moments. Acta Math. Hungar. 127, 64–84 (2010)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Szewczak Z.S.: A moment maximal inequality for dependent random variables. Statist. Probab. Lett. 106, 129–133 (2015)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Utev S.A.: Sums of random variables with φ-mixing. Siberian Adv. Math. 1, 124–155 (1991)MathSciNetzbMATHGoogle Scholar
  38. 38.
    D. Wajc, Negative association: definition, properties, and applications,

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland

Personalised recommendations