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Strong laws for the k-th order statistic when k≦c log2 n
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  • Published: April 1986

Strong laws for the k-th order statistic when k≦c log2 n

  • Paul Deheuvels1 

Probability Theory and Related Fields volume 72, pages 133–154 (1986)Cite this article

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Summary

Under general regularity assumptions, we characterize the upper and lower almost sure classes of U k, n , where U 1, n ...U n, n are the order statistics of an i.i.d. sample of size n from the uniform distribution on (0, 1), and where k=k n is a non-decreasing integer sequence such that 1≦k =O(log2 n) as n → ∞.

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References

  • Balkema, A.A., De Haan, L.: Limit distribution for order statistics I. Teoria Verjatnosti i Primenenia, 80–96 (1978)

  • Barndorff-Nielsen, O.: On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables. Math. Scand. 9, 383–394 (1961)

    Google Scholar 

  • Chung, K.L., Erdős, P.: On the application of the Borel-Cantelli lemma. Trans. Am. Math. Soc. 72, 179–186 (1952)

    Google Scholar 

  • Deheuvels, P.: Majoration et minoration presque sûre optimale des éléments de la statistique ordonnée d'un échantillon croissant de variables aléatoires indépendantes. Rendiconti della Classe di Science fisiche, mathematiche e naturali (Academia Nazionale dei Lincei) Serie 8, 6, 707–719 (1974)

    Google Scholar 

  • Deheuvels, P., Devroye, L.: Strong laws for the maximal k-spacing when k≦c log n. Z. Wahrscheinlichkeitstheor. Verw. Geb. 66, 315–334 (1984)

    Google Scholar 

  • Devroye, L.: Laws of the iterated logarithm for order statistics of uniform spacings. Ann. Probab. 9, 860–867 (1981)

    Google Scholar 

  • Devroye, L.: Upper and lower class sequences for minimal uniform spacings. Z. Wahrscheinlichkeitstheor. Verw. Geb. 61, 237–254 (1982)

    Google Scholar 

  • Geffroy, J.: Contributions à la théorie des valeurs extrêmes. Publications de l'Institut de Statistique des Universités de Paris 7–8, 37–185 (1958/1959)

    Google Scholar 

  • Hewitt, E., Savage, L.J.: Symmetric measures on cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955)

    Google Scholar 

  • Kiefer, J.: Iterated logarithm analogous for sample quantiles when p n ↓0. Proc. Sixth Berkeley Sympos. on Math. Statist. and Probab. 1, 227–244. University of California Press (1972)

    Google Scholar 

  • Robbins, H., Siegmund, D.: On the law of the iterated logarithm for maxima and minima. Proc. Sixth Berkeley Sympos. on Math. Statis. and Probab. 3, 51–70. University of California Press (1972)

    Google Scholar 

  • Shorack, G.R., Wellner, J.A.: Linear bounds on the empirical distribution function. Ann. Probab. 6, 349–353 (1978)

    Google Scholar 

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Authors and Affiliations

  1. L.S.T.A., Université de Paris VI, 4 Place Jussieu, F-75230, Paris Cedex 05, France

    Paul Deheuvels

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  1. Paul Deheuvels
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Deheuvels, P. Strong laws for the k-th order statistic when k≦c log2 n . Probab. Th. Rel. Fields 72, 133–154 (1986). https://doi.org/10.1007/BF00343900

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  • Received: 02 September 1985

  • Accepted: 05 December 1985

  • Issue Date: April 1986

  • DOI: https://doi.org/10.1007/BF00343900

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Keywords

  • Uniform Distribution
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Order Statistic
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