Sums of Independent Random Variables

  • Yuan Shih Chow
  • Henry Teicher
Part of the Springer Texts in Statistics book series (STS)


Of paramount concern in probability theory is the behavior of sums {S n , n ≥ 1} of independent random variables {X i, i ≥ 1}. The case where the {X i} are i.i.d. is of especial interest and frequently lends itself to more incisive results. The sequence of sums {S n, n ≥ 1} of i.i.d. r.v.s {X n} is alluded to as a random walk; in the particular case when the component r.v.s {X n} are nonnegative, the random walk is referred to as a renewal process.


Random Walk Independent Random Variable Stopping Time Uniform Integrability Especial Interest 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. J. H. Abbott and Y. S. Chow, “Some necessary conditions for a.s. convergence of sums of independent r.v.s.,” Bull. Institute Math. Academia Sinica 1 (1973), 1–7.MathSciNetzbMATHGoogle Scholar
  2. L. E. Baum and M. Katz, “Convergence rates in the law of large numbers,” Trans. Amer. Math. Soc. 120 (1965), 108–123.MathSciNetzbMATHCrossRefGoogle Scholar
  3. B. Brown, “Moments of a stopping rule related to the central limit theorem,” Ann. Math. Stat. 40 (1969), 1236–1249.zbMATHCrossRefGoogle Scholar
  4. D. L. Burkholder, “Independent sequences with the Stein property,” Ann. Math. Stat. 39(1968), 1282–1288.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Y. S. Chow, “Local convergence of martingales and the law of large numbers,” Ann. Math. Stat. 36 (1965), 552–558.zbMATHCrossRefGoogle Scholar
  6. Y. S. Chow, “Delayed sums and Borel summability of independent, identically distributed random variables,” Bull. Inst. Math., ACADEMIA SINICA 1 (1973), 207–220.zbMATHGoogle Scholar
  7. Y. S. Chow and H. Robbins, “On the asymptotic theory of fixed-width sequential confidence intervals for the mean.” Ann. Math. Stat. 36 (1965), 457–462.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Y. S. Chow and H. Teicher, “Almost certain summability of i.i.d. random variables,” Ann. Math. Stat. 42 (1971), 401–404.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Y. S. Chow, H. Robbins, and D. Siegmund, Great Expectations: The Theory of Optimal Stopping, Houghton Mifflin, Boston, 1972.Google Scholar
  10. Y. S. Chow, H. Robbins, and H. Teicher, “Moments of randomly stopped sums,” Ann. Math. Stat. 36 (1965), 789–799.MathSciNetzbMATHCrossRefGoogle Scholar
  11. K. L. Chung, “Note on some strong laws of large numbers,” Amer. Jour. Math. 69 (1947). 189–192.zbMATHCrossRefGoogle Scholar
  12. K. L. Chung, A Course in Probability Theory. Harcourt Brace, New York, 1968; 2nd ed., Academic Press, New York, 1974.Google Scholar
  13. K. L. Chung and W. H. J. Fuchs, “On the distribution of values of sums of random variables,” Mem. Amer. Math. Soe. 6 (1951).Google Scholar
  14. K. L. Chung and D. Ornstein, “On the recurrence of sums of random variables,” Bull. Amer. Math. Soe. 68 (1962), 30–32.MathSciNetzbMATHCrossRefGoogle Scholar
  15. C. Derman and H. Robbins, “The SLLN when the first moment does not exist,” Proe. Nat. Acad. Sei. U.S.A. 41 (1955), 586–587.MathSciNetzbMATHCrossRefGoogle Scholar
  16. J. L. Doob, Stochastic Processes, Wiley, New York, 1953.zbMATHGoogle Scholar
  17. K. B. Erickson, “The SLLN when the mean is undefined,” Trans. Amer. Math. Soc. 185 (1973), 371–381.MathSciNetCrossRefGoogle Scholar
  18. W. Feller, “Über das Gesetz der grossen Zahlen,” Acta Univ. Szeged., Sect. Sei. Math. 8 (1937), 191–201.zbMATHGoogle Scholar
  19. W. Feller, “A limit theorem for random variables with infinite moments,” Amer. Jour. Math. 68 (1946), 257–262.MathSciNetzbMATHCrossRefGoogle Scholar
  20. W. Feller, An Introduction to Probability Theory and its applications, Vol. 2, Wiley, New York, 1966.zbMATHGoogle Scholar
  21. R. Gundy and D. Siegmund, “On a stopping rule and the central limit theorem,” Ann. Math. Stat. 38 (1967), 1915–1917.MathSciNetzbMATHCrossRefGoogle Scholar
  22. C. C. Heyde, “Some renewal theorems with applications to a first passage problem,” Ann. Math. Stat. 37 (1966), 699–710.MathSciNetzbMATHCrossRefGoogle Scholar
  23. T. Kawata, Fourier Analysis in Probability Theory, Academic Press, New York, 1972.zbMATHGoogle Scholar
  24. H. Kesten, “The limit points of a random walk,” Ann. Math. Stat. 41 (1970), 1173–1205.MathSciNetzbMATHCrossRefGoogle Scholar
  25. A. Khintchine and A. Kolmogorov, “Über Konvergenz von Reichen, derem Glieder durch den Zufall bestimmt werden,” Ree. Math. (Mat. Sbornik) 32 (1924), 668–677.Google Scholar
  26. A. Kolmogorov, “Über die Summen durch den Zufall bestimmer unabhängiger Grössen,” Math. Ann. 99 (1928), 309–319; 102 (1930), 484–488.MathSciNetCrossRefGoogle Scholar
  27. M. J. Klass, “Properties of optimal extended-valued stopping rules,” Ann. Prob. 1 (1973), 719–757.MathSciNetzbMATHCrossRefGoogle Scholar
  28. M. Klass and H. Teicher, “Iterated logarithm laws for random variables barely with or without finite mean,” Ann. Prob. 5 (1977), 861–874.MathSciNetzbMATHCrossRefGoogle Scholar
  29. K. Knopp, Theory and Application of Infinite Series, Stechert-Hafner, New York, 1928.zbMATHGoogle Scholar
  30. P. Lévy, Theorie de l’addition des variables aléatoires, Gauthier-Villars, Paris, 1937; 2nd ed., 1954.Google Scholar
  31. M. Loève, “On almost sure convergence,” Proc. Second Berkeley Symp. Math. Stat. Prob., pp. 279–303, Univ. of California Press, 1951.Google Scholar
  32. M. Loève, Probability Theory, 3rd ed., Van Nostrand, Princeton, 1963; 4th ed., SpringerVerlag, Berlin and New York, 1977–1978.zbMATHGoogle Scholar
  33. J. Marcinkiewicz and A. Zygmund, “Sur les fonctions indépendantes,” Fund. Math. 29(1937), 60–90.Google Scholar
  34. P. Revesz, The Laws of Large Numbers, Academic Press, New York, 1968.zbMATHGoogle Scholar
  35. H. Robbins and E. Samuel, “An extension of a lemma of Wald,” J. Appl. Prob. 3 (1966), 272–273.MathSciNetzbMATHCrossRefGoogle Scholar
  36. F. Spitzer, “A combinatorial lemma and its applications to probability theory,” Trans. Amer. Math. Soc. 82 (1956), 323–339.MathSciNetzbMATHCrossRefGoogle Scholar
  37. H. Teicher, “Almost certain convergence in double arrays,” Z. Wahr. verw. Gebiete 69 (1985), 331–345.MathSciNetzbMATHCrossRefGoogle Scholar
  38. A. Wald, “On cumulative sums of random variables,” Ann. Math. Stat. 15 (1944), 283–296.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Yuan Shih Chow
    • 1
  • Henry Teicher
    • 2
  1. 1.Department of Mathematical StatisticsColumbia UniversityNew YorkUSA
  2. 2.Department of StatisticsRutgers UniversityNew BrunswickUSA

Personalised recommendations