### Abstract

For a given sequence of real numbers a_{1}, ..., a_{n}, we denote the kth smallest one by k-min_{1≤i≤n}a_{i}. Let A be a class of random variables satisfying certain distribution conditions (the class contains N(0,1) Gaussian random variables). We show that there exist two absolute positive constants c and C such that for every sequence of real numbers 0 < x_{1} ≤ ... ≤ x_{n} and every k ≤ n, one has c max_{1≤j≤k} k + 1 - j/∑_{i=j}^{n} 1/x _{i} ≤ double-struck E sign k- min_{1≤i≤n} |x _{i}ξ_{i}| ≤ C ln(k + 1) max_{1≤j≤k} k + 1 - j/∑_{i=j}^{n} 1/x_{i}, where ξ_{1}, ..., ξ_{n} are independent random variables from the class A. Moreover, if k = 1, then the left-hand side estimate does not require independence of the ξ_{i}'s. We provide similar estimates for the moments of k-min _{1≤i≤n}|x_{i}ξ_{i}| as well.

Original language | English (US) |
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Pages (from-to) | 3665-3675 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 134 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2006 |

### Keywords

- Expectations
- Exponential distribution
- Moments
- Normal distribution
- Order statistics

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## Cite this

*Proceedings of the American Mathematical Society*,

*134*(12), 3665-3675. https://doi.org/10.1090/S0002-9939-06-08453-X