Advertisement

Integration in a Probability Space

  • Yuan Shih Chow
  • Henry Teicher
Chapter
  • 1.7k Downloads
Part of the Springer Texts in Statistics book series (STS)

Abstract

There are two basic avenues to integration. In the modern approach the integral is introduced first for simple functions—as a weighted average of the values of the function—and then defined for any nonnegative measurable Function f as a limit of the integrals of simple nonnegative functions increasing to f. Conceptually this is extremely simple, but a certain price is paid in terms of proofs. The alternative classical approach, while employing a less intuitive definition, achieves considerable simplicity in proofs of elementary properties.

Keywords

Random Walk Probability Space Simple Random Walk Monotone Convergence Theorem Markov Inequality 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Y. S. Chow and W. J. Studden, “Monotonicity of the variance under truncation and variations of Jensen’s inequality,”Ann. Math. Stat.40 (1969), 1106–1108.MathSciNetzbMATHCrossRefGoogle Scholar
  2. K. L. Chung and W. H. J. Fuchs, “On the distribution of values of sums of random variables,”Mem. Amer. Soc.6 (1951).Google Scholar
  3. J. L. DoobStochastic ProcessesWiley, New York, 1953.zbMATHGoogle Scholar
  4. P. R. HalmosMeasure TheoryVan Nostrand, Princeton, 1950; Springer-Verlag, Berlin and New York, 1974.Google Scholar
  5. P. Hall, “On the 2pconvergence of random variables,”Proc. Cambridge Philos. Soc.82 (1977). 439–446.zbMATHCrossRefGoogle Scholar
  6. G. H. Hardy, J. E. Littlewood, and G. PolyaInequalitiesCambridge Univ. Press, London, 1934.Google Scholar
  7. S. B. Kochen and C. J. Stone, “A note on the Borel-Cantelli lemma,”Ill. Jour. Math.8 (1964), 248–251.MathSciNetzbMATHGoogle Scholar
  8. A. Liapounov, “ Nouvelle forme du théoreme sur la limite de probabilité,”Mem. Acad. Sc. St. Petersbourg12 (1905), No. 5.Google Scholar
  9. M. LoèveProbability Theory3rd ed., Van Nostrand, Princeton, 1963; 4th ed., Springer-Verlag, Berlin and New York, 1977–1978.Google Scholar
  10. G. Polya, “Uber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz,”Math. Ann.84 (1921), 149–160.MathSciNetzbMATHCrossRefGoogle Scholar
  11. S. SaksTheory of the Integral(L. C. Young, translation), Stechert-Hafner, New York, 1937.Google Scholar
  12. H. Teicher, “On the law of the iterated logarithm,”Ann. Prob.2 (1974), 714–728.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

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

Personalised recommendations