Acta Mathematica Hungarica

, Volume 49, Issue 1–2, pp 83–102 | Cite as

Almost sure invariance principles for the empirical process of lacunary sequences

  • S. Dhompongsa
Article

Keywords

Invariance Principle Empirical Process Lacunary Sequence 

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References

  1. [1]
    I. Berkes, An almost sure invariance principle for lacunary trigonometric series,Acta Math. Acad. Sci. Hungar.,26 (1975), 209–220.CrossRefGoogle Scholar
  2. [2]
    I. Berkes, On the asymptotic behaviour of Σf(n x x) I and II,Z. Wahrscheinlichkeitscheorie verw. Gebiete,41 (1977), 115–137.CrossRefGoogle Scholar
  3. [3]
    I. Berkes, On the central limit theorems of lacunary trigonometric series,Anal. Math.,4 (1978), 159–180.CrossRefGoogle Scholar
  4. [4]
    I. Berkes, A central limit theorem for trigonometric series with small gaps,Z. Wahrscheinlichkeitstheorie verw. Gebiete,47 (1979), 157–161.CrossRefGoogle Scholar
  5. [5]
    I. Berkes and W. Philipp, Approximation theorems for independent and weakly dependent random vectors,Ann. Probab.,7 (1979), 29–54.Google Scholar
  6. [6]
    I. Berkes and W. Philipp, An almost sure invariance principle for the empirical distribution function of mixing random variables,Z. Wahrscheinlichkeitstheorie verw. Gebiete,41 (1977), 115–137.CrossRefGoogle Scholar
  7. [7]
    Y. S. Chow and H. Teicher,Probability theory, Springer-Verlag (New York, Heidelberg, Berlin, 1978).Google Scholar
  8. [8]
    S. Dhompongsa, A note on the almost sure approximation of the empirical process of weakly dependent random vectors,Yokohama Mathematical Journal,32 (1984).Google Scholar
  9. [9]
    S. Dhompongsa,Limit theorems for weakly dependent random vectors Ph. D. Thesis, University of Illinois (Urbana, 1982).Google Scholar
  10. [10]
    S. Dhompongsa, Uniform laws of the iterated logarithm for Lipschitz classes of functions,Acta Sci. Math. (Szeged), (1986) (to appear).Google Scholar
  11. [11]
    V. G. Gaposhkin, Lacunary series and independent functions,Russian Math. Surveys,21 (1966), 3–82.Google Scholar
  12. [12]
    M. Kac, R. Salem and A. Zygmund, A gap theorem,Trans. Amer. Math. Soc.,63 (1948), 235–243.Google Scholar
  13. [13]
    M. B. Marcus and W. Philipp, Almost sure invariance principles for sums ofB-valued random variables with applications to random Fourier series and the empirical characteristic process. Preprint (1980).Google Scholar
  14. [14]
    W. Philipp, A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables,Ann. Probab.,5 (1977), 319–350.Google Scholar
  15. [15]
    W. Philipp and L. Pinzur, Almost sure approximation theorems for the multivariate empirical process,Z. Wahrscheinlichkeitstheorie verw. Gebiete,54 (1980), 1–13.CrossRefGoogle Scholar
  16. [16]
    W. F. Stout,Almost sure convergence, Academic Press (New York, 1974).Google Scholar
  17. [17]
    S. Takahashi, An asymptotic property of a gap sequence,Proc. Japan Acad.,38 (1962), 101–104.Google Scholar
  18. [18]
    S. Takahashi, The law of the iterated logarithm for a gap sequence with infinite gaps,Tôhoku Math. J.,15 (1963), 281–288.Google Scholar

Copyright information

© Akadémiai Kiadó 1987

Authors and Affiliations

  • S. Dhompongsa
    • 2
    • 1
  1. 1.Urbana
  2. 2.Chiang Mai UniversityChiang MaiThailand

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