Asymptotic expansions for sums of weakly dependent random vectors

  • F. Götze
  • C. Hipp


It is shown that formal Edgeworth expansions are valid for sums of weakly dependent random vectors. The error of approximation has ordero(n−(s−2)/2) if
  1. (i)

    the moments of orders+1 are uniformly bounded

  2. (ii)

    a conditional Cramér-condition holds

  3. (iii)

    the random vectors can be approximated by other random vectors which satisfy a strong mixing condition and a Markov type condition.


The strong mixing coefficients in (iii) are decreasing at an exponential rate. The above conditions can easily be checked and are often satisfied when the sequence of random vectors is a Gaussian, or a Markov, or an autoregressive process. Explicit formulas are given for the distribution of finite Fourier transforms of a strictly stationary time series.


Fourier Time Series Fourier Transform Stochastic Process Probability Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, T.W.: The Statistical Analysis of Time Series. New York: Wiley 1971Google Scholar
  2. 2.
    Bhattacharya, R.N., Ranga Rao, R.: Normal Approximation and Asymptotic Expansions. New York: Wiley 1976Google Scholar
  3. 3.
    Bolthausen, E.: The Berry-Essen theorem for functionals of discrete Markov chains. Z.Wahrscheinlichkeitstheorie verw. Gebiete54, 59–74 (1980)Google Scholar
  4. 4.
    Bulinskii, A.V., Zhurbenko, I.G.: The central limit theorem for random fields. Dokl. Akad. Nauk SSSR 226. English translation in: Soviet Math. Dokl.17, 14–17 (1976)Google Scholar
  5. 5.
    Durbin, J.: Approximations for densities of sufficient estimators. Biometrika67, 311–333 (1980)Google Scholar
  6. 6.
    Götze, F., Hipp, C.: Asymptotic expansions in the central limit theorem under moment conditions. Z. Wahrscheinlichkeitstheorie verw. Gebiete42, 67–87 (1978)Google Scholar
  7. 7.
    Hannan, E.J.: Multiple Time Series. New York: Wiley 1970Google Scholar
  8. 8.
    Ibragimov, I.A.: Some limit theorems for stationary processes. Theor. Probability Appl.7, 349–382 (1962)Google Scholar
  9. 9.
    Ibragimov, I.A.: The central limit theorem for sums of functions of independent random variables and sums of the form∑f(2 k t). Theor. Probability Appl.12, 596–607 (1967)Google Scholar
  10. 10.
    Ibragimov, I.A.: On the spectrum of stationary Gaussian sequences satisfying the strong mixing condition. II. Sufficient conditions. Mixing rate. Theor. Probability Appl.15, 23–36 (1970)Google Scholar
  11. 11.
    Rosenblatt, M.: A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA42, 43–47 (1956)Google Scholar
  12. 12.
    Statulevicius, V.: Limit theorems for sums of random variables that are connected in a Markov chain. I, II, III. Litovsk. Mat. Sb. 9, 345–362;9, 635–672 (1969);10, 161–169 (1970)Google Scholar
  13. 13.
    Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Sympos. Math. Statist. Probability2, 583–602 (1972)Google Scholar
  14. 14.
    Sweeting, T.J.: Speeds of convergence for the multidimensional central limit theorem. Ann. Probab.5, 28–41 (1977)Google Scholar
  15. 15.
    Tikhomirov, A.N.: On the convergence rate in the central limit theorem for weakly dependent random variables. Theor. Probability Appl.25, 790–809 (1980)Google Scholar
  16. 16.
    Zhurbenko, I.G.: On strong estimates of mixed semiinvariants of random processes. Sibirskii Mat. Z.13, 293–308. English translation in: Siberian Math. J.13, 202–213 (1972)Google Scholar
  17. 17.
    Grenander, U., Szegö, G.: Toeplitz Forms and Their Applications. Univ. California Press, Berkeley (1958)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • F. Götze
    • 1
  • C. Hipp
    • 1
  1. 1.Mathematisches Institut der Universität KölnKöln 41Federal Republic of Germany

Personalised recommendations