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On the rate of convergence of sums of extremes to a stable law
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  • Published: June 1990

On the rate of convergence of sums of extremes to a stable law

  • Arnold Janssen1 &
  • David M. Mason2 

Probability Theory and Related Fields volume 86, pages 253–264 (1990)Cite this article

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  • 4 Citations

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Summary

LetX 1,X 2, ..., be a sequence of independent and identically distributed random variables in the domain of normal attraction of a nonnormal stabler law. It is known that only the sum of thek n largest andk n smallest extreme values in thenth partial sum withk n →∞ andk n /n→0 are responsible for the asymptotic stable distribution of the whole sum. We investigate the rate at which such sums of extreme values converge to a stable law in conjunction with the rate at which the sums of the middle terms become asymptotically negligible. In terms of rates of convergence our results provide in many cases a quantitative measure of exactly what portion of the sample is asymptotically stable.

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References

  • Bennett, G.: Probability inequalities for the sum of independent random variables. J. Am. Stat. Assoc.57, 33–45 (1962)

    Google Scholar 

  • Bergström, H.: On some expansions of stable distribution functions. Ark. Mat.2, 375–378 (1952a)

    Google Scholar 

  • Bergström, H.: On distribution functions with a limiting stables distribution. Ark. Mat.2, 463–474 (1952b)

    Google Scholar 

  • Christoph, G.: Convergence rate in integral limit theorem with stable limit law. Lith. Math. J.19, 91–101 (1979)

    Google Scholar 

  • Christoph, G.: Über notwendige und hinreichende Bedingungen für Konvergenzgeschwindigkeitsaussagen im Falle einer stablien Grenzverteilung. Z. Wahrscheinlichkeitstheor. Verw. Geb.54, 29–40 (1980)

    Google Scholar 

  • Cramér, H.: On asymptotic expansions for sums of independent random variables with alimiting stable distribution. Sankya25, 13–24 (1963)

    Google Scholar 

  • Csörgő, S.: Notes on extreme and self-normalized sums from the domain of attraction of a stable law. J. London Math. Soc.39, 369–384 (1989)

    Google Scholar 

  • Csörgő, M., Csörgő, S., Horváth, L., Mason, D.M.: Normal and stable convergence of integral functions of the empirical distribution function. Ann. Probab.14, 86–118 (1986)

    Google Scholar 

  • Csörgő, S., Horváth, L., Mason, D.M.: What portion of the sample makes a partial sum asymptotically stable or normal? Probab. Th. Rel. Fields72, 1–16 (1986)

    Google Scholar 

  • Csörgő, S., Haeusler, E., Mason, D.M.: A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. Adv. Appl. Math.9, 259–333 (1988)

    Google Scholar 

  • Dubinskaite, J.: Accuracy of approximation of sums of independent random variables by a stable distribution. Lith. Math. J.23, 43–46 (1983)

    Google Scholar 

  • Feller, W.: An introduction to probability theory and its applications II. 2nd edn. New York: Wiley 1971

    Google Scholar 

  • Hall, P.: On the rate of convergence to a stable law. J. London Math. Soc. (2),23, 179–192 (1981a)

    Google Scholar 

  • Hall, P.: Two-si'ded bounds on the rate of convergence to a stable law. Z. Wahrscheinlichkeitstheor. Verw. Geb.57, 349–364 (1981b)

    Google Scholar 

  • Ibragimov, I.A., Linnik Yu, V.: Independent and stationary sequences of random variables. Groningen: Wolters-Noordhoff 1971

    Google Scholar 

  • Janssen, A.: The domain of attraction of stable laws and extreme order statistics. Probab. Math. Stat. (1989)

  • Mijnheer, J.: On the rate of convergence to a stable law. II. Lith. Math. J.26, 255–259 (1987)

    Google Scholar 

  • Mijnheer, J.: On the rate of convergence of the sum of the sample extremes. Probab. Th. Rel. Fields79, 317–325 (1988)

    Google Scholar 

  • Paulauskas, V.I.: Some nonuniform estimates in the limit theorems of probability theory. Sov. Math. Dokl.14, 1125–1127 (1973)

    Google Scholar 

  • Satyabaldina, K.I.: Absolute estimates of the rate of convergence to stable laws. Theory Probab. Appl.17, 726–728 (1972)

    Google Scholar 

  • Satyabaldina, K.I.: On the estimation of the rate of convergence in a limit theorem with a stable limit law. Theory Probab. Appl.18, 202–204 (1973)

    Google Scholar 

  • Zolotarev, V.M.: One dimensional stable distributions. Transl. Math. Monogr.,65, Providence (1986)

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Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Universität-GH-Siegen, Hölderlinstrasse 3, D-5900, Siegen, Federal Republic of Germany

    Arnold Janssen

  2. Department of Mathematical Sciences, University of Delaware, 19716, Newark, DE, USA

    David M. Mason

Authors
  1. Arnold Janssen
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  2. David M. Mason
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Additional information

Research partially supported by the Deutsche Forschungsgemeinschaft while visiting the University of Delaware

Research partially supported by NSF Grant no. DMS-8803209

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Janssen, A., Mason, D.M. On the rate of convergence of sums of extremes to a stable law. Probab. Th. Rel. Fields 86, 253–264 (1990). https://doi.org/10.1007/BF01474645

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  • Received: 03 March 1989

  • Revised: 05 January 1990

  • Issue Date: June 1990

  • DOI: https://doi.org/10.1007/BF01474645

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Keywords

  • Stochastic Process
  • Probability Theory
  • Quantitative Measure
  • Mathematical Biology
  • Stable Distribution
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