Random Sequences, Series, and Averages

  • Olav Kallenberg
Part of the Probability and Its Applications book series (PIA)


Convergence in probability and in LP; uniform integrabil ity and tightness; convergence in distribution; convergence of random series; strong laws of large numbers; Portmanteau the orem; continuous mapping and approximation; coupling and measurability.


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  1. The weak law of large numbers was first obtained by Bernoulli (1713) for the sequences named after him. More general versions were then established with increasing rigor by Bienaymé (1853), Chebyshev (1867), and Markov (1899). A necessary and sufficient condition for the weak law of large numbers was finally obtained by Kolmogorov (1928–29).Google Scholar
  2. Khinchin and Kolmogorov (1925) studied series of independent, discrete random variables and showed that convergence holds under the condition in Lemma 4.16. Kolmogorov (1928–29) then obtained his maximum inequality and showed that the three conditions in Theorem 4.18 are necessary and sufficient for a.s. convergence. The equivalence with convergence in distribution was later noted by Lévy (1954).Google Scholar
  3. The strong law of large numbers for Bernoulli sequences was stated by Borel (1909), but the first rigorous proof is due to Faber (1910). The simple criterion in Corollary 4.22 was obtained in Kolmogorov (1930). In (1933) kolmogorov showed that existence of the mean is necessary and sufficient for the strong law of large numbers for general i.i.d. sequences. The extension to exponents p ≠ 1 is due to Marcinkiewicz and Zygmund (1937). Proposition 4.24 was proved in stages by Glivenko (1933) and Cantelli (1933).Google Scholar
  4. Riesz (1909a) introduced the notion of convergence in measure, for probability measures equivalent to convergence in probability, and showed that it implies a.e. convergence along a subsequence. The weak compactness criterion in Lemma 4.13 is due to Dunford (1939). The functional representation of Proposition 4.31 appeared in Kallenberg (1996a), and Corollary 4.32 was given by Stricker and Yor (1978).Google Scholar
  5. The theory of weak convergence was founded by Alexandrov (1940–43), who proved in particular the so-called Portmanteau Theorem 4.25. The continuous mapping Theorem 4.27 was obtained for a single function f n = f by Mann and Wald (1943) and then in the general case by Prohorov (1956) and Rubin. The coupling Theorem 4.30 is due for complete S to Skorohod (1956) and in general to Dudley (1968).Google Scholar
  6. More detailed accounts of the material in this chapter may be found in many textbooks, such as in Loève (1977) and Chow and Teicher (1997). Additional results on random series and a.s. convergence appear in Stout (1974) and Kwapien and Woyczynski (1992).Google Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Olav Kallenberg
    • 1
  1. 1.Department of MathematicsAuburn UniversityAuburnUSA

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