Joint Characteristic Functions and Associated Sequences

  • André Robert Dabrowski
Part of the Progress in Probability and Statistics book series (PRPR, volume 11)


The proof of a limit theorem for independent and identically distributed random variables usually uses independence in a fundamental way. For dependent sequences, the dependence structure is frequently used to approximate the classical tools of independence, i.e. the expression of joint probabilities as a product of marginal probabilities, and/or the factorization of joint characteristic functions. Here, we wish to review the connection between characteristic functions, associated sequences and two fundamental limit theorems.


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  1. [1]
    Barlow, R. and F. Proschan (1975). Statistical Theory of Reliability and Life Testing. New York. Holt, Rinehard and Winston.Google Scholar
  2. [2]
    Berkes, I. (1973). The Functional Low of the Iterated Logarithm for Dependent Random Variables. Z. Wahrsch. verw. Gebiete 26; 245–258.Google Scholar
  3. [3]
    Berkes, I. (1973). On Strassen’s Version of the Loglog Law for Multiplicative Systems. Studia Sci. Math. Hung. 8, 425–431.Google Scholar
  4. [4]
    Berkes, I. (1973). Functional Limit Theorems for Lacunary Trigonmetric and Walsh Series. Studia Sci. Math. Hung. 8, 411–423.Google Scholar
  5. [5]
    Billingsley, P. (1968). Convergence of Probability Measures. Wiley. New York.zbMATHGoogle Scholar
  6. [6]
    Cox, J.T. and G. Grimmett (1984). Central Limit Theorems for Associated Random Variables and the Percolation Model. A. Probability 12, 514–528.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    Dabrowski, A.R. (1985). A Functional Law of the Iterated Logarithm for Associated Sequences. To appear in Statistics and Probability Letters.Google Scholar
  8. [8]
    Donsker, M. (1951). An Invariance Principle for Certain Probability Limit Theorems. Mem. Amer. Math. Soc. 6.Google Scholar
  9. [9]
    Esary, J., F. Proschan and D. Walkup (1967). Association of Random Variables with Applications. Ann. Math Statist. 38, 1466–1474.Google Scholar
  10. [10]
    Fortuin, C.P. Kastelyn and J. Ginibre (1971). Correlation Inequalities on some Partially Ordered Sets. Comm. Math. Phys. 22, 89–103.Google Scholar
  11. [11]
    Harris, T.E. (1960). A Lower Bound for the Critical Probability in a Certain Percolation Process. Proc. Camb. Phil. Soc. 59, 13–20.Google Scholar
  12. [12]
    Herrndorf, N. (1984). An Example on the Central Limit Theorem for Associated Sequences. A. Probability 12, 912–917.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    Lai, T.L. (1977), Power-one Tests Based on Sample Sums. A. Statistics, 5, 866–880.CrossRefzbMATHGoogle Scholar
  14. [14]
    Lebowitz, J. (1972). Rounds on the Correlations and Analyticity Properties of Ferromagnetic Ising Spin Systems. Comm. Math. Phys. 28, 313–321.Google Scholar
  15. [15]
    Lehmann, E.L. (1966). Some Concepts of Dependence. Ann. Math. Statist. 37, 1137–1153.Google Scholar
  16. [16]
    Newman, C.M. (1980). Normal Functuations and the FKG Inequalities. Comm. Math. Phys. 74, 119–128.Google Scholar
  17. [17]
    Newman. C.M. (1983). A General Central Limit Theorem for FKG Systems. Comm. Math. Phys. 91, 75–80.Google Scholar
  18. [18]
    Newman, C.M. (1984). Asymptotic Independence and Limit Theorems for Positively and Negatively Dependent Random Variables. Inequalities in Statistics and Probability. Volume 5 of the I:?S Notes-Monograph Series, Y.L. Tong, editor.Google Scholar
  19. [19]
    Newman, C.M. and A.L. Wright (1981). An Invariance Principle for Certain Dependent Sequences. A. Probability 9, 671–675.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    Newman, C.M. and A.L. Wright (1982). Associated Random Variables and Martingale Inequalities. Z. Wahrsch. verw. Gebiete. 59, 361–371.Google Scholar
  21. [21]
    Oodaira, H. (1975). Some Functional Laws of the Iterated Logarithm for Dependent Random Variables. Collo. Math. Soc. Janos Bolyai 11, 289–299.Google Scholar
  22. [22]
    Philipp, W. and W. Stout (1975). Almost Sure Invariance Principles for Partial sums of Weakly Dependent random variables. Mem. Amer. Math. Soc. 161.Google Scholar
  23. [23]
    Pitt, L. (1982). Positively Correlated Normal Variables are Associated. A. Probability, 10, 496–499.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    Robbins, H. (1970). Statistical Methods Related to the Law of Iterated Logarithm. Ann. Math. Statist. 41, 1397–1409.Google Scholar
  25. [25]
    Robbins, H. and D. Siegmund (1974). Expected Sample Size of some Tests of Power One. A. Statistics 2, 415–436.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [26]
    Rosenblatt, M. (1956). A Central Limit Theorem and a Strong Mixing Condition. Proc. Natl. Acad. Sci. USA 42, 43–47.Google Scholar
  27. [27]
    Strassen, V. (1964). An Invariance Principle for the Law of the Iterated Logarithm. Z. Wahrsch. verw. Gebiete 3, 211–226.Google Scholar
  28. [28]
    Wood, T.E. (1983). A Berry-Essen Theorem for Associated Random Variables. A. Probability 11, 1042–1047.Google Scholar
  29. [29]
    Wood, T.E. Associated (1983). A Berry-Esseen Theorem for Random Variables. A. Probability 11, (1985). A Local Limit Theorem for Sequences. A Probability 13, Vol. 2.Google Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • André Robert Dabrowski
    • 1
  1. 1.Department of MathematicsUniversity of OttawaOttawaCanada

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