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Sums, Trimmed Sums and Extremes pp 355–376Cite as

On the Asymptotic Behavior of Sums of Order Statistics from a Distribution with a Slowly Varying Upper Tail

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Part of the book series: Progress in Probability ((PRPR,volume 23))

Abstract

Let (X n)n≥1 be a sequence of independent non-negative random variables from a common distribution function F with a regiilarly varying upper tail. A number of results are presented on the stability, asymptotic distribution and law of the iterated logarithm for trimmed sums formed by deleting a number of the upper extreme values from the partial sum X 1 +...+X n at each stage n. The methods of proof are entirely based on quantile function techniques. This paper should provide the reader with a good introduction to some of the possibilities and the scope of this methodology.

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

Research partially supported by the Alexander von Humboldt Foundation while the author was visiting the University of Munich and NSF Grant DMS-8803209.

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References

  1. BINGHAM, N.H., GOLDIE, C.M. and TEUGELS, J.L. (1987). Regular Variation. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  2. CSAKI, E. (1977). The law of the iterated logarithm for normalized empirical distribution function. Z. Wahrsch. Verw, Gebiete 38 147–167.

    Article  MathSciNet  MATH  Google Scholar 

  3. CSÖRGÖ, S., HAEUSLER, E. and MASON, D.M. (1988a). A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. Adv. in Appl. Math. 9 259–333.

    Article  MathSciNet  Google Scholar 

  4. CSÖRGÖ, S., HAEUSLER, E. and MASON, D.M. (1988b). Asymptotic distribution of trimmed sums. Ann. Probab. 16 672–699.

    Article  MathSciNet  Google Scholar 

  5. DARLING, D.A. (1952). The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 95–107.

    Article  MathSciNet  MATH  Google Scholar 

  6. EINMAHL, J.H.J, and MASON, D.M. (1988). Laws of the iterated logarithm in the tails for weighted uniform empirical processes. Ann. Probab. 16 126–141.

    Article  MathSciNet  MATH  Google Scholar 

  7. Einmahl, J.H.J., HAEUSLER, E., and MASON, D.M. (1988). On the relationship between the almost sure stability of weighted empirical distributions and sums of order statistics. Prob. Th. Rel. Fields 79 59–74.

    Article  MathSciNet  MATH  Google Scholar 

  8. HAAN, DE, L. (1975). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tracts 32, Amsterdam.

    Google Scholar 

  9. HAEUSLER, E. and MASON, D.M. (1987). Laws of the iterated logarithm for the middle portion of the sample. Math. Proc. Cambridge Philos. Soc. 101 301–312

    Article  MathSciNet  Google Scholar 

  10. KIEFER, J. (1972). Iterated logarithm analogues for sample quantities when Pn ↓ 0. Proc. Sixth Berkeley Symp. Math. Statist. Prohab. Vol. I, 227–244.

    Google Scholar 

  11. MALLER, R.A. and RESNICK, S.L (1984). Limiting behaviour of sums and the term of maximum modulus. Proc. London Math. Soc. 49 385–422.

    Article  MathSciNet  MATH  Google Scholar 

  12. MIKOSCH, T. (1988). Interated logarithm results for rapidly growing random walks. Statistics 19 107–115.

    Article  MathSciNet  MATH  Google Scholar 

  13. MORI, T. (1976). The strong law of large numbers when extreme terms are excluded from sums. Z. Wahrsch. Verw. Gebiete 36, 189–194.

    Article  MATH  Google Scholar 

  14. MORI, T. (1981). The relation of sums and extremes of random variables. Session Summary Booklet, Invited Papers, Buenos Aires Session, Nov. 30-Dec. 11, (International Statistical Institute) 879–894.

    Google Scholar 

  15. PRUITT, W.E. (1981). General one-sided laws of the iterated logarithm.Ann. Probab. 9 1–48.

    Article  MathSciNet  MATH  Google Scholar 

  16. PRUITT, W.E. (1987). The contribution to the sum of the summand of maximum modulus. Ann. Probab. 15 885–896.

    Article  MathSciNet  MATH  Google Scholar 

  17. REVESZ, P. and WILLEKENS, E. (1987). On the maximal distance between two renewal epochs. Stoc. Proc. 27 21–41.

    Article  MathSciNet  MATH  Google Scholar 

  18. SHORACK, G.R. (1980). Some laws of the iterated logarithm type results from the empirical process. Austral. J. Statist. 22 50–59.

    Article  MathSciNet  MATH  Google Scholar 

  19. SHORACK, G.R. and WELLNER, J.A. (1982). Limit theorems and inequalities for the uniform empirical process indexed by intervals. Ann. Probab. 10, 639–652.

    Article  MathSciNet  MATH  Google Scholar 

  20. SHORACK, G.R. and WELLNER, J.A. (1986). Empirical Processes with Application to Statistics. Wiley, New York.

    Google Scholar 

  21. STIGLER, S.M. (1973). The asymptotic distribution of the trimmed mean. Ann. Statist. 1 472–477.

    Article  MathSciNet  MATH  Google Scholar 

  22. STUTE, W. (1982). The oscillation behavior of empirical processes. Ann. Probab. 10 86–107.

    Article  MathSciNet  MATH  Google Scholar 

  23. TEICHER, H. (1979). Rapidly growing random walks and a associated stopping time. Ann. Probab. 7 1078–1081.

    Article  MathSciNet  MATH  Google Scholar 

  24. WELLNER, J.A. (1978). Limit theorems for the ratio of the empirical distribution function to the true distribution function. Z. Wahrsch. Verw. Gebiete 45 73–88.

    Article  MathSciNet  MATH  Google Scholar 

  25. ZHANG, C.-H. (1986). The lower limit of a normalized random walk.Ann. Probab. 14 560–581.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Institute, University of Munich, Theresienstrasse 39, 8000, Munich 2, West Germany

    Erich Haeusler

  2. Department of Mathematics Sciences, University of Delaware, 19716, Newark, Delaware, USA

    David M. Mason

Authors
  1. Erich Haeusler
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  2. David M. Mason
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Tufts University, 02155, Medford, MA, USA

    Marjorie G. Hahn

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

    David M. Mason

  3. Department of Mathematics, Boston University, 02215, Boston, MA, USA

    Daniel C. Weiner

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© 1991 Birkhäuser Boston

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Haeusler, E., Mason, D.M. (1991). On the Asymptotic Behavior of Sums of Order Statistics from a Distribution with a Slowly Varying Upper Tail. In: Hahn, M.G., Mason, D.M., Weiner, D.C. (eds) Sums, Trimmed Sums and Extremes. Progress in Probability, vol 23. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-6793-2_12

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  • DOI: https://doi.org/10.1007/978-1-4684-6793-2_12

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4684-6795-6

  • Online ISBN: 978-1-4684-6793-2

  • eBook Packages: Springer Book Archive

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