Advertisement

Probability-1 pp 373-460 | Cite as

Convergence of Probability Measures. Central Limit Theorem

  • Albert N. Shiryaev
Part of the Graduate Texts in Mathematics book series (GTM, volume 95)

Abstract

In the formal construction of a course in the theory of probability, limit theorems appear as a kind of superstructure over elementary chapters, in which all problems have finite, purely arithmetical character. In reality, however, the epistemological value of the theory of probability is revealed only by limit theorems. Moreover, without limit theorems it is impossible to understand the real content of the primary concept of all our sciences – the concept of probability.

References

  1. [9]
    P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.zbMATHGoogle Scholar
  2. [15]
    A. V. Bulinsky and A. N. Shiryayev. Theory of Random Processes [Teoriya Sluchaĭnykh Processov]. Fizmatlit, Moscow, 2005.Google Scholar
  3. [29]
    C.-G. Esseen. A moment inequality with an application to the central limit theorem. Skand. Aktuarietidskr., 39 (1956), 160–170.Google Scholar
  4. [34]
    B. V. Gnedenko and A. N. Kolmogorov. Limit Distributions for Sums of Independent Random Variables, revised edition. Addison-Wesley, Reading, MA, 1968.zbMATHGoogle Scholar
  5. [39]
    P. R. Halmos. Measure Theory. Van Nostrand, New York, 1950.CrossRefzbMATHGoogle Scholar
  6. [45]
    E. V. Khmaladze, Martingale approach in the theory of nonparametric goodness-of-fit tests. Probability Theory and its Applications, 26, 2 (1981), 240–257.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [48]
    A. Kolmogoroff, Sulla determinazione empirica di una leggi di distribuzione. Giornale dell’Istituto degli Attuari, IV (1933), 83–91.Google Scholar
  8. [52]
    A. N. Kolmogorov and S. V. Fomin. Elements of the Theory of Functions and Functionals Analysis. Graylok, Rochester, 1957 (vol. 1), 1961 (vol. 2); sixth edition [Elementy teorii funktsiĭ ifunktsionalnogo analiza]. “Nauka” Moscow, 1989.Google Scholar
  9. [55]
    Ky Fan, Entfernung zweier zufälliger Grössen and die Konvergenz nach Wahrscheinlichkeit. Math. Zeitschr. 49, 681–683.Google Scholar
  10. [57]
    L. Le Cam. An approximation theorem for the Poisson binomial distribution. Pacif. J. Math., 19, 3 (1956), 1181–1197.MathSciNetzbMATHGoogle Scholar
  11. [63]
    R. Sh. Liptser and A. N. Shiryaev. Theory of Martingales. Kluwer, Dordrecht, Boston, 1989.CrossRefzbMATHGoogle Scholar
  12. [75]
    Yu. V. Prohorov [Prokhorov]. Asymptotic behavior of the binomial distribution. Uspekhi Mat. Nauk 8, no. 3(55) (1953), 135–142 (in Russian).Google Scholar
  13. [76]
    Yu. V. Prohorov. Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1 (1956), 157–214.MathSciNetCrossRefGoogle Scholar
  14. [82]
    V. I. Rotar. An extension of the Lindeberg–Feller theorem. Math. Notes 18 (1975), 660–663.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [87]
    I. G. Shevtsova. On absolute constants in the Berry-Esseen inequality and its structural and nonuniform refinements. Informatika i ee primeneniya. 7, 1 (2013), 124–125 (in Russian).Google Scholar
  16. [94]
    N. V. Smirnov. On the deviations of the empirical distribution curve [Ob ukloneniyakh empiricheskoĭ krivoĭ raspredeleniya] Matem. Sbornik, 6, (48), no. 1 (1939), 3–24.Google Scholar
  17. [99]
    V. M. Zolotarev. Modern Theory of Summation of Random Variables [Sovremennaya Teoriya Summirovaniya Nezavisimyh Sluchaĭnyh Velichin]. Nauka, Moscow, 1986.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Albert N. Shiryaev
    • 1
  1. 1.Department of Probability Theory and Mathematical StatisticsSteklov Mathematical Institute and Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations