Abstract
A new proof of Gnedenko’s theorem for the convergence to the Frêchet extreme distributions is presneted. The proof makes use of the theory of stable laws on R+.
Uniform rates of convergence are obtained. The paper highlights the role of mixtures of exponential distributions in extreme value theory.
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References
Bingham N.H. and Goldie C.M.: (1982a) Extensions of regular variation, I. Uniformity and quantifiers. Proc. London Math. Soc.
Bingham N.H. and Goldie C.M.: (1982b) Extensions of regular variation, II. Representation and indices. Proc. London Math. Soc.
Cohen J.P.: (1982) Convergence rates for the ultimate and penultimate approximations in extreme value theory. Adv. Appl. Prob. 14, 833–854.
Feller W.: (1971) An introduction to Probability theoryand its Applications. Vol 2, 2nd ed. Wiley, New-York.
Galambos J.: (1978) The asymptotic theory of extreme order statistics. Wiley, New-York.
Gomes I.: (1984) Penultimate limiting forms in extreme value theory. Ann.Inst.Stat. Math. Vol 36, 1, 71–84.
Hall P.: (1981) Two sided bounds on the rate of convergence to a stable law. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 57, 349–369.
Mori T: (1981) The relations of sums and extremes of random variables. Proc.43 rd. Session I.S.I. Buenos-Aires.
Seneta E: (1976) Regularly varying functions. Lecture Notes in Mathematics 508, Springer New-York.
Smith R.L. (1982) Uniform rates of convergence in extreme value theory. Adv. Appl. Prob. 14, 600–622.
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© 1987 D. Reidel Publishing Company
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Broniatowski, M. (1987). Convergence for the Sample Extremes Via Convolutions. In: Puri, M.L., Révész, P., Wertz, W. (eds) Mathematical Statistics and Probability Theory. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3963-9_5
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DOI: https://doi.org/10.1007/978-94-009-3963-9_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8258-7
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