Uniform estimates for order statistics and Orlicz functions

Yehoram Gordon, Alexander E. Litvak, Carsten Schütt, Elisabeth Werner

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12 Scopus citations

Abstract

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ 1,..., ξ n and a vector of scalars x = (x 1,..., x n), and 1 ≤ k ≤ n, we provide estimates for E κ-min1≤ κ ≤ n {pipe}x iξ{pipe} and E κ-min1≤ κ ≤ n {pipe}x iξ{pipe} in terms of the values κ and the Orlicz norm {double pipe} y x {double pipe} of the vector y x = (1/x 1,..., 1/x n). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable {pipe}ξ 1{pipe}, G(t)= ℙ({{pipe}ξ{pipe}}) For example, if ξ 1 is the standard N(0, 1) Gaussian random variable, then,. We would like to emphasize that our estimates do not depend on the length n of the sequence.

Original languageEnglish (US)
Pages (from-to)1-28
Number of pages28
JournalPositivity
Volume16
Issue number1
DOIs
StatePublished - Mar 2012
Externally publishedYes

Bibliographical note

Funding Information:
Y. Gordon, A. E. Litvak, C. Schütt and E. Werner were partially supported by AIM, Palo Alto. Y. Gordon was supported in part by “France-Israel Cooperation Grant #3-1350” and by the “Fund for the Promotion of Research at the Technion”. E. Werner was partially supported by a NSF Grant, by a FRG Grant, and by a BSF Grant.

Keywords

  • Expectations
  • Exponential distribution
  • Moments
  • Normal distribution
  • Order statistics
  • Orlicz norms

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