Recent Advances in the Central Limit Theorem and Its Weak Invariance Principle for Mixing Sequences of Random Variables (A Survey)

  • Magda Peligrad
Part of the Progress in Probability and Statistics book series (PRPR, volume 11)


The purpose of this paper is to describe the progress that has recently been made in the study of the central limit theorem and its weak invariance principle for mixing sequences of random variables and to point out some open problems in this subject.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berbee, H. C. P., (1979); Random Walks With Stationary Increments and Renewal Theory. Mathematical Centre, Amsterdam.zbMATHGoogle Scholar
  2. 2.
    Boldrighini, C., De Massi, A., Nogueira, A., Presutti, E., (1985); The Dynamics of a particle interacting with a semi-infinite ideal gas in a Bernoulli flow. Statistical Physics and Dynamical Systems (Progress in Physics) Birkhäuser, p. 153.Google Scholar
  3. 3.
    Bradley, R. C. (1983), Equivalent Measures of Dependence, J. Multivariate Analysis 13, 167–176.CrossRefzbMATHGoogle Scholar
  4. 4.
    Bradley, R. C. (1980), A Remark on the Central Limit Question for Dependent Random Variances, J. Appl. Prob. 17, 94–101.CrossRefzbMATHGoogle Scholar
  5. 5.
    Bradley, R. C. (1981 ), A Sufficient Condition for Linear Growth of Variables in a Stationary Random Sequence, Proc. Amer. Math. Soc. 83, 586–589.Google Scholar
  6. 6.
    Bradley, R. C. (1981 ), Central Limit Theorems under Weak Dependence, J. Multivariate Analysis, 11, 1–16.Google Scholar
  7. 7.
    Bradley, R. C. (1984 ), On the Central Limit Question under Absolute Regularity (to appear in Ann. of Prob.).Google Scholar
  8. 8.
    Bradley, R. C. ( 1984 ), The Central Limit Question under a-mixing (to appear in Rocky Mtn. J. Math.).Google Scholar
  9. 9.
    Bradley, R. C. (1983 ), Information Regularity and Central Limit Question, Rocky Mtn. J. Math., 13, 77–97.Google Scholar
  10. 10.
    Bradley, R. C., Bryc, W. (1985), Multilinear Forms and Measures of Dependence Between Random Variables, J. Multivariate Analysis, 16, 335–367.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Bradley, R. C. (1982), Counterexamples to the Central Limit Theorem Under Strong Mixing Conditions, Colloquia Mathematica Soc. Jénos Bolyai. 36 Limit Theorems in Probability and Statistics, Veszprém (Hungary).Google Scholar
  12. 12.
    Bradley, R. C. (1985), The Basic Properties of Strong Mixing Conditions, (to appear in Dependence in Probability and Statistics, Birkhauser).Google Scholar
  13. 13.
    Bradley, R. C., Peligrad, M. (1985), Invariance Principles Under A Two-Part Mixing Assumption. Preprint.Google Scholar
  14. 14.
    Bryc, W. (1985), Central Limit Theorem for 4-mixing Random Sequences Under Lindeberg-type Condition, (preprint).Google Scholar
  15. 15.
    Cogburn, R. (1960), Asymptotic Properties of Stationary Sequences, Univ. Calif. Publ. Statist. 3, 99–146.Google Scholar
  16. 16.
    Davydov, Y. A. (1968), Convergence of Distributions Generated by Stationary Stochastic Processes., Teor. Probability Appl. 13, 691–696.CrossRefzbMATHGoogle Scholar
  17. 17.
    Davydov, Y. A. (1969), On the Strong Mixing Property for Markov Chains with a Countable Number of States, Soviet Math. Doki. 10, 825–827.zbMATHGoogle Scholar
  18. 18.
    Davydov, Y. A. (1973), Mixing Conditions for Markov Chains, Theory Probab. Appl. 18, 312–328.zbMATHGoogle Scholar
  19. 19.
    Dehling, H., Denker, M., Philipp, W., (1984), Central Limit Theorems for Mixing Sequences of Random Variables under Minimal Conditions (to appear in Ann. Probability).Google Scholar
  20. 20.
    Dehling, H. and Philipp, W. (1982), Almost Sure Invariance Principle for Weakly Dependent Vector-Valued Random Variables, Ann. Prob. 10, 689–701.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Denker, M. (1978), A Note on the Central Limit Theorem for Strongly Mixing Random Variables. Preprint.Google Scholar
  22. 22.
    Denker, M. (1979), A Limit Theorem for Mixing Stationary Processes and its Applications. Preprint.Google Scholar
  23. 23.
    Denker, M. (1985), Uniform Integrability and the Central Limit Theorem (to appear in Dependence in Probability and Statistics. Birkhäuser).Google Scholar
  24. 24.
    Dvoretzky, A. (1972), Asymptotic Normality for Sums of Dependent Random Variables, Proc. Sixth Berkeley Sympos. Math. Stat. Prob., Vol. 2, 515–535.Google Scholar
  25. 25.
    Eberlein, E. (1984), On Strong Invariance Principle Under Dependence Assumptions, (to appear in Ann. Probability).Google Scholar
  26. 26.
    Gastwirth, J. L., Rubin, H. (1975), The Asymptotic Distribution Theory of the Empiric c.d.f. for Mixing Stochastic Processes, Ann. Statist., 3, 809–824.CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Hall, P., Heyde, C. C. (1980), Martingale Limit Theory and Its Applications, Academic Press, New York.Google Scholar
  28. 28.
    Hahn, M. G., Kuelbs, J. (1985), Asymptotic Normality of Trimmed Sums of + Mixing Random Variables. Preprint.Google Scholar
  29. 29.
    Herrndorf, N. (1983 ), The Invariance Principle for +-mixing Sequences, Z. Wahrsch. Verw. Gebiete 63, 1, 97–109.Google Scholar
  30. 30.
    Herrndorf, N., (1983 ), Stationary Strongly Mixing Sequences Not Satisfying the Central Limit Theorem, Annals Prob. 11, 809–813.Google Scholar
  31. 31.
    Herrndorf, N., ( 1983 ), A Functional Central Limit Theorem for Strongly Mixing Sequences of Random Variables, (Z. Wahrsch. verw. Gebiete, submitted).Google Scholar
  32. 32.
    Herrndorf, N., (1984 ), A Functional Central Limit Theorem for P-Mixing Sequences, J. of Multiv. Anal. 15, 141–146.Google Scholar
  33. 33.
    Herrndorf, N., (1984 ), A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables, Ann. of Prob., 12, 141–153.Google Scholar
  34. 34.
    Hoffman-Jorgensen, J., (1974), Sums of Independent Banach Space Valued Random Variables, Studia Math. 52, 159–186.MathSciNetGoogle Scholar
  35. 35.
    Ibragimov, I. A., (1962), Some Limit Theorem for Stationary Processes, Theory Prob. Appl. 7, 349–382.Google Scholar
  36. 36.
    Ibragimov, I. A., (1975), A Note on the Central Limit Theorem for Dependent Random Variables, Theory Prob. Appl. 20, 135–141.zbMATHGoogle Scholar
  37. 37.
    Ibragimov, I. A., Linnik, Y. V. (1971), Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.zbMATHGoogle Scholar
  38. 38.
    Ibragimov, I. A., Rozanov, Y. A., (1978), Gaussian Random Processes, Springer-Verlag, Berlin.CrossRefzbMATHGoogle Scholar
  39. 39.
    Iosifescu, M. (1980), Recent Advances in Mixing Sequences of Random Variables. Third International Summer School on Probability Theory and Mathematical Statistics, Varna. 1978.Google Scholar
  40. 40.
    Iosifescu, M., Teodorescu, R. (1969), Random Processes and Learning, Springer-Verlag, New York.CrossRefzbMATHGoogle Scholar
  41. 41.
    Jakubowski, A. (1985), A Note on the Invariance Principle for Stationary 4-Mixing Sequences. Tightness via Stopping Times. Preprint.Google Scholar
  42. 42.
    Kesten, H., O’Brien, G. L. (1976), Examples of Mixing Sequences, Duke Math. J. 43, 405–415.zbMATHMathSciNetGoogle Scholar
  43. 43.
    Kolmogorov, A. N., Rozanov, Y. A. (1960), On Strong Mixing Conditions for Stationary Gaussian Processes. Theory Probab. Appl. 5, 204–208.MathSciNetGoogle Scholar
  44. 44.
    Lai, T. L. (1977), Convergence Rates and r-Quick Version of the Strong Law for Stationary Mixing Sequences, Ann. Prob. 5, 693–706.CrossRefzbMATHGoogle Scholar
  45. 45.
    Lai, T. L., Robbins, H. (1978), A Class of Dependent Random Variables and Their Maxima, Z. Wahrsch. Verw. Gebiete 42, 89–111.CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    McLeish, D. L. (1975), Invariance Principles for Dependent Variables, Z. Wahrsch. verw. Gebiete 32, 165–178.CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    McLeish, D. L. (1977), On the Invariance Principles for Nonstationary Mixingales, Ann. Prob. 5, 616–621.CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Moricz, F. (1982), A General Moment Inequality for the Maximum of Partial Sums of Single Series, Acta Sci. Math. 44, 67–75.zbMATHMathSciNetGoogle Scholar
  49. 49.
    Moricz, F., Serfling, R. J. and Stout, W. (1982), Moment and Probability Bounds with Quasi-Superadditive Structure for the Maximum Partial Sum, Ann. Prob. 10, 1032–1040.CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    Nahapetian, B. S. (1984), On Limit Theorem for Dependent Random Variables. The Sixth International Symposium on Information Theory. September 18–22, 1984, Tashkent.Google Scholar
  51. 51.
    Oodaira, H. and Yoshihara, K. (1971), The Law of the iterated Logarithm for Stationary Processes Satisfying Mixing Conditions, Kodai Math. Sem. Rep. 23, 311–334.CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Oodaira, H. and Yoshihara, K. (1972), Functional Central Limit Theorems for Strictly Stationary Processes Satisfying the Strong Mixing Condition, Kodai Math. Sem. Rep. 24, 259–269.CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    Peligrad, M. (1981 ), An Invariance Principle for Dependent Random Variables, Z. Wahrsch. verw. Gebiete 57, 495–507.Google Scholar
  54. 54.
    Peligrad, M. ( 1981 ), A Maximal Inequality for 4-Mixing Sequences, Center of Math. Statistics Notes, Bucharest.Google Scholar
  55. 55.
    Peligrad, M. (1982), Invariance Principles for Mixing Sequences of Random Variables, The Ann. of Prob. 10, 4, 968–981.CrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    Peligrad, M. (1983), A Note on Two Measures of Dependence and Mixing Sequences, Adv. Appl. Prob. 15, 461–464.CrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    Peligrad, M. (1984 ), An Invariance Principle for 4-Mixing Sequences, (to appear in Ann. of Prob.).Google Scholar
  58. 58.
    Peligrad, M. (1984 ), Invariance Principles under Weak Dependence (to appear in J. of Multivariate Analysis).Google Scholar
  59. 59.
    Philipp, W. (1979), An Almost Sure Invariance Principle for Sums of B-Valued Random Variables, Lecture Notes In Math. 709, 171–193, Springer-Verlag, New York.Google Scholar
  60. 60.
    Philipp, W. (1980), Weak and Lu-Invariance Principle for Sums of B-Valued Random Variables, The Ann. of Prob. 8, 1, 68–82.CrossRefMathSciNetGoogle Scholar
  61. 61.
    Philipp, W., Stout, W. F. (1975), Almost Sure Invariance Principles for Sums of Weakly Dependent Random Variables, Amer. Math. Soc. Memoir, No. 161.Google Scholar
  62. 62.
    Philipp, W., Webb, G. R. (1973), An Invariance Principle for Mixing Sequences of Random Variables, Z. Warhsch. verw. Gebiete 25, 223–237.CrossRefzbMATHMathSciNetGoogle Scholar
  63. 63.
    Rosenblatt, M. (1956), A Central Limit Theorem and a Strong Mixing Condition, Proc. Nat. Acad. Sci. USA 42, 43–47.CrossRefzbMATHMathSciNetGoogle Scholar
  64. 64.
    Rosenblatt, M. (1971), Markov Processes, Structure and Asymptotic Behavior, Springer-Verlag, Berlin.CrossRefGoogle Scholar
  65. 65.
    SamAr, J. D 19x4), ectorsgeitn stationarry Rows, Ann. robar12, rrays of an om Vectors w 390–426.Google Scholar
  66. 66.
    Serfling, R. J. (1968), Contributions to Central Limit Theory for Dependent Variables, Ann. Math. Stat. 39, 1158–1175.Google Scholar
  67. 67.
    Shields, P., (1973), The Theory of Bernoulli Shifts. University of Chicago Press, Chicago.zbMATHGoogle Scholar
  68. 68.
    Simons, G., Stout, W. (1978), A Weak Invariance Principle with Applications to Domains of Attraction, Ann. Prob. 6, 294–315.CrossRefzbMATHMathSciNetGoogle Scholar
  69. 69.
    Volkonskii, V. A., Rozanov, Y. A. (1959), Some Limit Theorems for Random Functions I. Theory Prob. Appl. 4, 178–197.MathSciNetGoogle Scholar
  70. 70.
    Yokoyama, R. (1980), Moment Bounds for Stationary Mixing Sequences, Z. Wahrsch. verw. Gebiete 52, 45–57.CrossRefzbMATHMathSciNetGoogle Scholar
  71. 71.
    Yoshihara, K. (1978), Probability Inequalities for Sums of Absolutely Regular Processes and Their Applications, Z. Wahrsch. verw. Gebiete 43, 319–329.CrossRefzbMATHMathSciNetGoogle Scholar
  72. 72.
    Withers, C. L. (1981), Central Limit Theorems for Dependent Variables I, Z. Wahrsch. verw. Gebiete 57, 509–534.CrossRefzbMATHMathSciNetGoogle Scholar
  73. 73.
    Withers, C. L. (1983), Corringendum to Central Limit Theorems for Dependent Random Variables, Z. Wahrsch. verw. Gebiete, 63, 555.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • Magda Peligrad
    • 1
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

Personalised recommendations