Advertisement

Basic Properties of Strong Mixing Conditions

  • Richard C. Bradley
Part of the Progress in Probability and Statistics book series (PRPR, volume 11)

Abstract

This is a survey of the basic properties of strong mixing conditions for sequences of random variables. The focus will be on the “structural” properties of these conditions, and not at all on limit theory. For a discussion of central limit theorems and related results under these conditions, the reader is referred to Peligrad [60] or Iosifescu [50]. This survey will be divided into eight sections, as follows:
  1. 1.

    Measures of dependence

     
  2. 2.

    Five strong mixing conditions

     
  3. 3.

    Mixing conditions for two or more sequences

     
  4. 4.

    Mixing conditions for Markov chains

     
  5. 5.

    Mixing conditions for Gaussian sequences

     
  6. 6.

    Some other special examples

     
  7. 7.

    The behavior of the dependence coefficients

     
  8. 8.

    Approximation of mixing sequences by other random sequences.

     

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Berbee, Random walks with stationary increments and renewal theory. Mathematical Centre, Amsterdam, 1979.zbMATHGoogle Scholar
  2. [2]
    H. Berbee. Weak Bernoulli processes, periodicity and renewal theory on the integers. Report no. 167. Mathematical Centre, Amsterdam, 1981.Google Scholar
  3. [3]
    H. Berbee. Convergence rates in the strong law for bounded mixing sequences. Report MS-R8412. Mathematical Centre, Amsterdam, 1984.Google Scholar
  4. [4]
    H. Berbee and R.C. Bradley. A limitation of Markov representation for stationary processes. Stochastic Process. Appl. 18 (1984) 33–45.zbMATHMathSciNetGoogle Scholar
  5. [5]
    J. Bergh and J. Löfström, Interpolation spaces. Springer-Verlag, Berlin, 1976.CrossRefzbMATHGoogle Scholar
  6. [6]
    I. Berkes and W. Philipp. An almost sure invariance principle for the empirical distribution function of mixing random variables. Z. Wahrsch. Verw, Gebiete 41 (1977) 115–137.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    I. Berkes and W. Philipp. Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 (1979)29–54.Google Scholar
  8. [8]
    J.R. Blum, D.L. Hanson, and L.H. Koopmans. On the strong law of large numbers for a class of stochastic processes, Z. Wahrseh. Verw. Gebiete 2 (1963) 1–11.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics 470. Springer-Verlag, Berlin, 1975.Google Scholar
  10. [10]
    R. Bowen. Bernoulli maps of the interval. Israel J. Math. 28 (1977) 161–168.zbMATHMathSciNetGoogle Scholar
  11. [11]
    R.C. Bradley. On the (1)-mixing condition for stationary random sequences. Duke Math. J. 47 (1980) 421–433.zbMATHGoogle Scholar
  12. [12]
    R.C. Bradley. Central limit theorems under weak dependence. J. Multivariate Anal. 11 (1981) 1–16Google Scholar
  13. [13]
    R.C. Bradley. Absolute regularity and functions of Markov chains. Stochastic Process, Appl. 14 (1983) 67–77Google Scholar
  14. [14]
    R.C. Bradley. On the 1p-mixing condition for stationary random sequences. Trans. Amer. Math. Soc. 276 (1983) 55–66.zbMATHMathSciNetGoogle Scholar
  15. [15]
    R.C. Bradley. Equivalent measures of dependence. J. Multivariate Anal, 13 (1983) 167–176.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    R.C. Bradley. Approximation theorems for strongly mixing random variables, Michigan Math, J. 30 (1983) 69–81.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    R.C. Bradley. On a very weak Bernoulli condition, Stochastics 13 (1984) 61–81.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    R.C. Bradley. Identical mixing rates. Preprint, 1984.. (Sub- mitted for publication.)Google Scholar
  19. [19]
    R.C. Bradley. A bilaterally deterministic p-mixing stationary random sequence. Trans. Amer. Math. Soc. (1986), to appear.Google Scholar
  20. [20]
    R.C. Bradley. On the central limit question under absolute regularity. Ann. Probab. (1985), to appear.Google Scholar
  21. [21]
    R.C. Bradley and W. Bryc. Multilinear forms and measures of dependence between random variables. J. Multivariate Anal. 16 (1985) 335–367.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    R.C. Bradley, W. Bryc, and S. Janson. Some remarks on the foundations of measures of dependence. Tech. Rept. No. 105, Center for Stochastic Processes, University of North Carolina, Chapel Hill, 1985.Google Scholar
  23. [23]
    W. Bryc. On the approximation theorem of I. Berkes and W. Philipp. Demonstratio Math. 15 (1982) 807–816.zbMATHMathSciNetGoogle Scholar
  24. [24]
    A.V. Bulinskii. On measures of dependence close to the maximal correlation coefficient. Soviet Math. Dokl. 30 (1984) 249–252.Google Scholar
  25. [25]
    R. Cogburn. Asymptotic properties of stationary sequences. Univ. Calif. Publ. Statist. 3 (1960) 99–146.MathSciNetGoogle Scholar
  26. [26]
    H. Cohn. On a class of dependent random variables. Rev. Roumanie Math. Pures Appl. 10 (1965) 1593–1606.zbMATHGoogle Scholar
  27. [27]
    P. Csaki and J. Fischer. On the general notion of maximal correlation. Magyar Tud. Akad. Mat. Kutato Int. Kozl. 8 (1963) 27–51.zbMATHMathSciNetGoogle Scholar
  28. [28]
    A.R. Dabrowski. A note on a theorem of Berkes and Philipp for dependent sequences. Statist. Probab. Letters 1 (1982) 53–55.CrossRefGoogle Scholar
  29. [29]
    R.A. Davis. Stable limits for partial sums of dependent random variables. Ann. Probab. 11 (1983) 262–269.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [30]
    Y.A. Davydov. Mixing conditions for Markov chains. Theory Probab. Appl. 18 (1973) 312–328.zbMATHGoogle Scholar
  31. [31]
    H. Dehling. A note on a theorem of Berkes and Philipp. Z. Wahrsch. Verw. Gebiete 62 (1983) 39–42.CrossRefGoogle Scholar
  32. [32]
    H. Dehling, M. Denker, and W. Philipp. Versik processes and very weak Bernoulli processes with summable rates are independent. Proc. Amer. Math. Soc. 91 (1984) 618–624.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    H. Dehling and W. Philipp. Almost sure invariance principles for weakly dependent vector-valued random variables. Ann. Probab. 10 (1982) 689–701.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [34]
    M. Denker and G. Keller. On U-statistics and V. Mises’ statistics for weakly dependent processes. Z. Wahrsch. V.rw. Gebiete 64 (1983) 505–522.CrossRefzbMATHMathSciNetGoogle Scholar
  35. [35]
    M. Denker and W. Philipp. Approximation by Brownian motion for Gibbs measures and flows under a function. Ergodic Theory Dynamical Systems 4 (1984) 541–552.CrossRefzbMATHMathSciNetGoogle Scholar
  36. [36]
    E. Eberlein. Strong approximation of very weak Bernoulli processes, Z. Wahrsch. Verw. Gebiete 62 (1983) 17–37.CrossRefGoogle Scholar
  37. [37]
    E. Eberlein. Weak convergence of partial sums of absolutely regular sequences. Statist. Probab. Letters 2 (1984) 291–293.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [38]
    M. Falk. On the convergence of spectral densities of arrays of weakly stationary processes. Ann. Probab. 12 (1984) 918–921.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [39]
    H. Gebelein. Das Statistische Problem der Korrelation als Variations-und Eigenwertproblem and sein Zusammenhang mit der Ausgleichungsrechnung. Z. Angew. Math. Mech. 21 (1941) 364–379.Google Scholar
  40. [40]
    M.I. Gordin. The central limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969) 1174–1176.zbMATHGoogle Scholar
  41. [41]
    D. Griffeath. A maximal coupling for Markov chains. Z. Wahrsch. Verw. Gebiete 31 (1975) 95–106.CrossRefzbMATHMathSciNetGoogle Scholar
  42. [42]
    P. Hall and C.C. Heyde. Martingale Limit Theory and Its Application. Academic Press, New York, 1980.zbMATHGoogle Scholar
  43. [43]
    N. Herrndorf. Stationary strongly mixing sequences not satisfying the central limit theorem. Ann. Probab. 11 (1983) 809–813.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [44]
    H.O. Hirschfeld. A connection between correlation and contingency. Proc. Camb. Phil. Soc. 31 (1935) 520–524.Google Scholar
  45. [45]
    F. Hofbauer and G. Keller. Ergodic properties in invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982) 119–140.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [46]
    I.A. Ibragimov. Some limit theorems for strictly stationary stochastic processes. Dokl. Akad. Nauk SSSR 125 (1959) 711–714. (In Russian).Google Scholar
  47. [47]
    I.A. Ibragimov and Y.V. Linnik. Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen, 1971.zbMATHGoogle Scholar
  48. [48]
    I.A. Ibragimov and Y.A. Rozanov. Gaussian Random Processes. Nauka, Moscow, 1970. (In Russian).Google Scholar
  49. [49]
    I.A. Ibragimov and Y.A. Rozanov. Gaussian Random Processes. Springer-Verlag, Berlin, 1978.CrossRefzbMATHGoogle Scholar
  50. [50]
    M. Iosifescu. Recent advances in mixing sequences of random variables. In: Third international summer school on probability theory and mathematical statistics, Varna 1978, pp. 111–138. Publishing House of the Bulgarian Academy of Sciences, Sofia, 1980.Google Scholar
  51. [51]
    H. Kesten and G.L. O’Brien. Examples of mixing sequences. Duke Math. J. 43 (1976) 405–415.CrossRefzbMATHMathSciNetGoogle Scholar
  52. [52]
    A.N. Kolmogorov and Y.A. Rozanov. On strong mixing conditions for stationary Gaussian processes. Theory Probab. Appl. 5 (1960) 204–208.MathSciNetGoogle Scholar
  53. [53]
    A. Lasota and J.A. Yorke. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973) 481–488.CrossRefMathSciNetGoogle Scholar
  54. [54]
    M.R. Leadbetter, G. Lindgren, and H. Rootzen. Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, Berlin, 1983.CrossRefzbMATHGoogle Scholar
  55. [55]
    F. Ledrappier. Some properties of absolute continuous invariant measures on an interval. Ergodic Theory Dynamical Systems 1 (1981) 77–93.CrossRefzbMATHMathSciNetGoogle Scholar
  56. [56]
    B.A. Lifshits. Invariance principle for weakly dependent variables. Theory Probab. Appl. 29 (1984) 33–40.MathSciNetGoogle Scholar
  57. [57]
    S. Orey. Lecture notes on limit theorems for Markov chain transition probabilities. Mathematical Studies 34. Van Nostrand Reinhold, New York, 1971.Google Scholar
  58. [58]
    M. Peligrad. Invariance principles for mixing sequences of random variables. Ann. Probab. 10 (1982) 968–981.CrossRefzbMATHMathSciNetGoogle Scholar
  59. [59]
    M. Peligrad. A note on two measures of dependence and mixing sequences. Adv. in Appl. Probab. 15 (1983) 461–464.CrossRefMathSciNetGoogle Scholar
  60. [60]
    M. Peligrad. Recent advances in the central limit theorem and its invariance principle for mixing sequences of random variables. This Birkhäuser volume.Google Scholar
  61. [61]
    T.D. Pham and L.T. Tran. Some mixing properties of time series models. Stochastic Process. Appl. 19 (1985) 297–303.zbMATHMathSciNetGoogle Scholar
  62. [62]
    W. Philipp. Some metrical theorems in number theory II. Duke Math. J. 37 (1970) 447–458.zbMATHMathSciNetGoogle Scholar
  63. [63]
    W. Philipp. Mixing sequences of random variables and probabilistic number theory. Mem. Amer. Math. Soc. No. 114, 1971.Google Scholar
  64. [64]
    W. Philipp and W. Stout. Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. No. 161, 1975.Google Scholar
  65. [65]
    M.S. Pinsker, Information and Information Stability of Random Variables and Processes. Holden-Day, San Francisco, 1964.zbMATHGoogle Scholar
  66. [66]
    M. Rosenblatt. A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. U.S.A. 42 (1956) 43–47.CrossRefGoogle Scholar
  67. [67]
    M. Rosenblatt, Markov Processes, Structure and Asymptotic Behavior. Springer-Verlag, Berlin, 1971.CrossRefzbMATHGoogle Scholar
  68. [68]
    M. Rosenblatt, Uniform ergodicity and strong mixing. Z. Wahrsch. Verw. Gebiete 24 (1972) 79–84.CrossRefzbMATHGoogle Scholar
  69. [69]
    M. Rosenblatt. Linear processes and bispectra. J. Appl, Probab. 17 (1980) 265–270.Google Scholar
  70. [70]
    D. Sarason. An addendum to ‘Past and Future’. Math. Scand. 30 (1972) 62–64.MathSciNetGoogle Scholar
  71. [71]
    P. Shields. The Theory of Bernoulli Shifts. Univ. of Chicago Press, Chicago, 1973.zbMATHGoogle Scholar
  72. [72]
    E.M. Stein and G. Weiss. An extension of a theorem of Marcinkiewicz and some of its applications. J. Math. Mech. 8 (1959) 263–284.zbMATHMathSciNetGoogle Scholar
  73. [73]
    V.A. Volkonskii and Y.A. Rozanov. Some limit theorems for random functions I. Theory Probab. Appl. 4 (1959) 178–197.CrossRefMathSciNetGoogle Scholar
  74. [74]
    V.A. Volkonskii and Y.A. Rozanov. Some limit theorems for random functions II. Theory Probab. Appl. 6 (1961) 186–198.CrossRefGoogle Scholar
  75. [75]
    C.S. Withers. Central limit theorems for dependent random variables I. Z. Wahrsch. Verw. Gebiete 57 (1981) 509–534.CrossRefzbMATHMathSciNetGoogle Scholar
  76. [76]
    H.S. Witsenhausen. On sequences of pairs of dependent random variables. SIAM J. Appl. Math. 28 (1975) 100–113.zbMATHMathSciNetGoogle Scholar
  77. [77]
    A. Zygmund. Trigonometric Series, Vol. I. Cambridge Univ. Press, Cambridge, 1959.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • Richard C. Bradley
    • 1
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA

Personalised recommendations