## 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 _{i}ξ{pipe} and E κ-min1≤ κ ≤ n {pipe}x _{i}ξ{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 language | English (US) |
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Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Positivity |

Volume | 16 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2012 |

Externally published | Yes |

### 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