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Almost sure and weak invariance principles for random variables attracted by a stable law
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  • Published: September 1989

Almost sure and weak invariance principles for random variables attracted by a stable law

  • I. Berkes1 &
  • H. Dehling2 

Probability Theory and Related Fields volume 83, pages 331–353 (1989)Cite this article

  • 79 Accesses

  • 4 Citations

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Summary

Let (X n ) be i.i.d. random variables belonging to the domain of normal attraction of a symmetric stable law with parameter 0<p<2. We study the a.s. and weak approximation of the partial sum process\(S(t) - \sum\limits_{n \leqq t} {X_n (t \geqq } 0)\) by a symmetric stable processG p(t). Stout proved an upper bound for the optimal remainder term in this approximation; we prove here a lower bound, leaving only a small gap between the upper and lower estimates. We also give a new method to obtain upper bounds. Finally, we prove analogues of these results in the case when a.s. approximation is replaced by approximation in probability.

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References

  1. deAcosta, A., Giné, E.: Convergence of moments and related functionals in the general central limit theorem in Banach spaces. Z. Wahrscheinlichkeitstheor. Verw. Geb.48, 213–231 (1979)

    Google Scholar 

  2. Berkes, I., Dabrowski, A., Dehling, H., Philipp, W.: A strong approximation theorem for sums of random vectors in the domain of attraction to a stable law. Acta. Math. Hungar.48, 161–172 (1986)

    Google Scholar 

  3. Cramér, H.: On the approximation to a stable probability distribution. In: Szegö, G. (ed.) Studies in mathematical analysis and related topics (Essays in honor of G. Pólya), pp. 70–76. Stanford, Calif.: Stanford University Press 1962

    Google Scholar 

  4. Feller, W.: An introduction to probability theory and its applications, vol. II. New York: Wiley 1966

    Google Scholar 

  5. Fisher, E.: An almost sure invariance principle for random variables in the domain of attraction of a stable law. Z. Wahrscheinlichkeitstheor. Verw. Geb.67, 461–471 (1984)

    Google Scholar 

  6. Heyde, C.C.: On large deviation problems for sums of random variables which are not attracted by the normal law. Ann. Math. Statist.38, 1575–1578 (1967)

    Google Scholar 

  7. Lévy, P.: Théorie de l'addition des variables aléatoires. Paris: Gauthiers-Villars 1937

    Google Scholar 

  8. Loève, M.: Probability theory. 2nd ed. New York: Van Nostrand 1963

    Google Scholar 

  9. Major, P.: An improvement of Strassen's invariance principle. Ann. Probab.7, 55–61 (1979)

    Google Scholar 

  10. Mijnheer, J.: Strong approximations of partial sums of i.i.d. random variables in the domain of attraction of a symmetric stable distribution. Trans. of the 9th Prague conference on information theory, stat. decision functions, random processes. Prague: Academia 1983

    Google Scholar 

  11. Seneta, E.: Regularly varying functions. (Lect. Notes. Math., vol. 508) Berlin Heidelberg New York: Springer 1976

    Google Scholar 

  12. Simons, G., Stout, W.F.: A weak invariance principle with applications to domains of attraction. Ann. Probab.6, 294–315 (1978)

    Google Scholar 

  13. Stout, W.F.: Almost sure invariance principles whenEX 21 =∞. Z. Wahrscheinlichkeitstheor. Verw. Geb.49, 23–32 (1979)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, Reáltanoda u. 13-15, H-1053, Budapest, Hungary

    I. Berkes

  2. Department of Mathematics and Computing Science, Groningen University, P.O. Box 800, NL-9700 AV, Groningen, The Netherlands

    H. Dehling

Authors
  1. I. Berkes
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  2. H. Dehling
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Additional information

Research supported by Hungarian National Foundation for Scientific Research, grant no. 18081 f(x)~g(x),f(x)\(<< \) g(x),f(x)≈g(x)(x→∞) mean, as usual,\(\mathop {\lim }\limits_{x \to \infty } f(x)/g(x) = 1\),\(\mathop {\overline {\lim } }\limits_{x \to \infty } |f(x)/g(x)|\infty \),\(0< \mathop {\lim }\limits_{x \to \infty } f(x)/g(x)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \mathop {\overline {\lim } }\limits_{x \to \infty } f(x)/g(x)< + \infty \), respectively

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Berkes, I., Dehling, H. Almost sure and weak invariance principles for random variables attracted by a stable law. Probab. Th. Rel. Fields 83, 331–353 (1989). https://doi.org/10.1007/BF00964369

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  • Received: 28 January 1986

  • Revised: 17 October 1988

  • Issue Date: September 1989

  • DOI: https://doi.org/10.1007/BF00964369

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Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Lower Estimate
  • Invariance Principle
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