Skip to main content

Limit Theorems for Independent Random Variables

  • Chapter
Probability Theory

Part of the book series: Springer Texts in Statistics ((STS))

  • 2884 Accesses

Abstract

Prior discussion of the strong and weak laws of large numbers centered around the i.i.d. case. Necessary and sufficient conditions for the weak law are available when the underlying random variables are merely independent and have recently been obtained for the strong law as well. Unfortunately, the practicality of the latter conditions leaves much to be desired.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • L. E. Baum and M. Katz, “Convergence rates in the law of large numbers,”Trans. Amer. Math. Soc. 120(1965), 108–123.

    Article  MathSciNet  MATH  Google Scholar 

  • H. D. Brunk, “The strong law of large numbers,”Duke Math. J. 15(1948), 181–195.

    Article  MathSciNet  MATH  Google Scholar 

  • Y. S. Chow, “A martingale inequality and the law of large numbers,”Proc. Amer. Math. Soc. 11(1960), 107–111.

    Article  MathSciNet  MATH  Google Scholar 

  • Y. S. Chow, “On a strong law of large numbers for martingales,”Ann. Math. Stat. 38(1967),610.

    Article  MATH  Google Scholar 

  • Y. S. Chow, “ Delayed sums and Borel summability of independent, identically distributed random variablesBull. Inst. Math. Academia Sinica 1(1973), 207–220.

    MATH  Google Scholar 

  • Y. S. Chow and T. L. Lai, “Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossingsTrans. Amer. Math. Soc.(1975).

    Google Scholar 

  • Y. S. Chow, H. Robbins, and D. SiegmundGreat Expectations: The Theory of Optimal StoppingHoughton Mifflin, Boston, 1972.

    Google Scholar 

  • K. L. Chung, “Note on some strong laws of large numbers,”Amer. J. Math. 69(1947), 189–192.

    Article  MathSciNet  MATH  Google Scholar 

  • K. L. Chung, “The strong law of large numbers,”Proc. 2nd Berkeley Symp. Stat. and Prob.(1951), 341–352.

    Google Scholar 

  • J. L. DoobStochastic ProcessesWiley, New York, 1953.

    MATH  Google Scholar 

  • V. A. Egorov, “On the strong law of large numbers and the law of the iterated logarithm for sequences of independent random variables.”Theor. Prob. Appl. 15(1970), 509–514.

    Article  Google Scholar 

  • W. FellerAn Introduction to Probability Theory and Its ApplicationsVol. 2, Wiley, New York, 1966.

    MATH  Google Scholar 

  • W. Feller, “An extension of the law of the iterated logarithm to variables without variance,”Journ. Math. and Mech. 18(1968), 343–356.

    MathSciNet  MATH  Google Scholar 

  • B. V. Gnedenko and A. N. KolmogorovLimit Distributions for Sums of Independent Random Variables(K. L. Chung, translator), Addison-Wesley, Reading, Mass., 1954.

    Google Scholar 

  • P. Hartman and A. Wintner, “On the law of the iterated logarithm,”Amer. Jour. Math.63 (1941), 169–176.

    Article  MathSciNet  Google Scholar 

  • P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbersProc. Nat. Acad. Sci. U.S.A.33 (1947), 25–31.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Khintchine, “Über dyadische Bruche,”Math. Zeit. 18(1923), 109–116.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Khintchine, “Über einen Satz der Wahrscheinlichkeitsrechnung,”Fund. Math. 6 (1924)9–20.

    Google Scholar 

  • J. Kiefer and J. Wolfowitz, “On the characteristics of the general queueing process with applications to random walk,”Ann. Math. Stat. 27(1956), 147–161.

    Article  MathSciNet  MATH  Google Scholar 

  • J. F. C. Kingman, “Some inequalities for the queue G1/G/1,”Biometrika 49(1962), 315–324.

    MathSciNet  MATH  Google Scholar 

  • A. Kolmogorov, “Über der Gesetz des Iterierten Logarithmus,”Math. Annalen 101(1929), 126–135.

    Article  Google Scholar 

  • M. LoèveProbability Theory3rd ed., Van Nostrand, Princeton, 1963; 4th ed., Springer-Verlag, Berlin and New York, 1977–1978.

    Google Scholar 

  • J. Marcinkievicz and A. Zygmund, “Sur les fonctions indépendantes,”Fund. Math. 29(1937), 60–90.

    Google Scholar 

  • J. Marcinkiewicz and A. Zygmund, “Quelques théorèmes sur les fonctions indépendantes,”Studia Math. 7(1938), 104–120.

    Google Scholar 

  • S. V. Nagaev, “On necessary and sufficient conditions for the strong law of large numbers,”Theor. Prob. Appl. 17 (1972)573–581.

    Article  MathSciNet  MATH  Google Scholar 

  • Y. V. Prohorov. “The strong law of large numbers,”lzv. Akad. Nauk. Ser. Mat. 14(1950), 523–536 [in Russian].

    MathSciNet  Google Scholar 

  • Y. V. Prohorov, “Some remarks on the strong law of large numbers,”Theor. Prob. Appl. 4(1959), 201–208.

    Article  MathSciNet  Google Scholar 

  • D. A. Raikov, “On a connection between the central limit theorem in the theory of probability and the law of large numbers,”Izv. Nauk USSR Soy. Math.(1938), 323–338.

    Google Scholar 

  • D. Siegmund, “On moments of the maximum of normed sums,”Ann. Math. Stat. 40(1969), 527–531.

    Article  MathSciNet  MATH  Google Scholar 

  • F. Spitzer, “A combinatorial lemma and its applications to probability theory,”Trans. Amer. Math. Soc. 82(1956), 323–339.

    Article  MathSciNet  MATH  Google Scholar 

  • F. Spitzer, “The Wiener-Hopf equation whose kernal is a probability density,”Duke Math. Jour. 24(1957), 327–343.

    Article  MathSciNet  MATH  Google Scholar 

  • V. Strassen, “A converse to law of the iterated logarithm,”Z. Wahr. 4 (1966) 265–268.

    Article  MathSciNet  MATH  Google Scholar 

  • H. M. Taylor, “Bounds for stopped partial sums,”Ann. Math. Stat. 43 (1972) 733–747.

    Article  MATH  Google Scholar 

  • H. Teicher, “A dominated ergodic type theorem,”Z. Wahr. 8 (1967) 113–116.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Teicher, “Some new conditions for the strong law,”Proc. Nat. Acad. Sci. U.S.A. 59 (1968) 705–707.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Teicher, “Completion of a dominated ergodic theorem,”Ann. Math. Stat. 42 (1971) 2156–2158.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Teicher, “On interchangeable random variables,”Studi di Probabilità Statistica e Ricerca Operative in Onore di Giuseppe Pompilj pp. 141–148Gubbio1971.

    Google Scholar 

  • H. Teicher, “On the law of the iterated logarithm” Ann. Prob. 2 (1974) 714–728.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Teicher, “A necessary condition for the iterated logarithm law,”Z. Wahr. Verw. Geb. (1975) 343–349.

    Google Scholar 

  • H. Teicher, “Generalized exponential bounds, iterated logarithm and strong laws,”Z. Wahr. Verw. Geb. (1979) 293–307.

    Google Scholar 

  • N. Wiener, “The ergodic theorem,”Duke Math. Jour. 5 (1938) 1–18.

    Article  MathSciNet  Google Scholar 

  • A. ZygmundTrigonometric SeriesVol. I, Cambridge1959.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Science+Business Media New York

About this chapter

Cite this chapter

Chow, Y.S., Teicher, H. (1997). Limit Theorems for Independent Random Variables. In: Probability Theory. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1950-7_10

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1950-7_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-40607-7

  • Online ISBN: 978-1-4612-1950-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics