## Abstract

For a given sequence of real numbers a_{1},..., a_{n} we denote the k-th smallest one by k-min_{1≤i≤n}a_{i}. We show that there exist two absolute positive constants c and C such that for every sequence of positive real numbers x_{1},..., x_{n} and every k ≤ n one has c max_{1≤j≤k} k+1-j/∑_{i=j}^{n} 1/x_{i} ≤ E k- min_{1≤i≤n} x_{i}g_{i} ≤ C ln(k + 1) max_{1≤j≤k} k+1-j/∑_{i=j}^{n} 1/x_{i}, where g_{i} ∈ N(0, 1), i = 1,..., n, are independent Gaussian random variables. Moreover, if k = 1 then the left hand side estimate does not require independence of the g_{i}s. Similar estimates hold for E k- min_{1≤i≤n} x_{i} g_{i} ^{p} as well.

Original language | English (US) |
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Pages (from-to) | 445-448 |

Number of pages | 4 |

Journal | Comptes Rendus Mathematique |

Volume | 340 |

Issue number | 6 |

DOIs | |

State | Published - Mar 15 2005 |

Externally published | Yes |

### Bibliographical note

Funding Information:E-mail addresses: gordon@techunix.technion.ac.il (Y. Gordon), alexandr@math.ualberta.ca (A. Litvak), schuett@math.uni-kiel.de (C. Schütt), emw2@po.cwru.edu (E. Werner). 1 This author is partially supported by the Fund for the Promotion of Research at the Technion. 2 This author is partially supported by FP6 Marie Curie Actions, MRTN-CT-2004-511953, PHD. 3 This author is partially supported by a NSF Grant, by a Nato Collaborative Linkage Grant and by a NSF Advance Opportunity Grant.