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Proceedings of the American Mathematical Society

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On the minimum of several random variables

Authors: Y. Gordon, A. E. Litvak, C. Schütt and E. Werner
Journal: Proc. Amer. Math. Soc. 134 (2006), 3665-3675
MSC (2000): Primary 62G30, 60E15, 60G51
Published electronically: May 31, 2006
MathSciNet review: 2240681
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Abstract: For a given sequence of real numbers $ a_{1}, \dots, a_{n}$, we denote the $ k$th smallest one by $ {k\mbox{-}\min} _{1\leq i\leq n}a_{i}$. Let $ \mathcal{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}\leq \ldots \leq x_{n}$ and every $ k\leq n$, one has

$\displaystyle c \max_{1 \leq j \leq k} \frac {k+1-j}{\sum_{i=j}^n 1/x_i } \leq \mathbb{E} \, \, k$-$\displaystyle \min_{1\leq i\leq n} \vert x_{i} \xi_{i}\vert \leq C\, \ln(k+1)\, \max_{1 \leq j \leq k} \frac{k+1-j}{\sum_{i=j}^n 1/x_i}, $

where $ \xi_1, \dots, \xi_n$ are independent random variables from the class $ \mathcal{A}$. Moreover, if $ k=1$, then the left-hand side estimate does not require independence of the $ \xi_i$'s. We provide similar estimates for the moments of $ {k\mbox{-}\min}_{1\leq i\leq n} \vert x_{i} \xi_{i}\vert$ as well.

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Additional Information

Y. Gordon
Affiliation: Department of Mathematics, Technion, Haifa 32000, Israel

A. E. Litvak
Affiliation: Department of Mathematics and Statistics Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

C. Schütt
Affiliation: Mathematisches Seminar, Christian Albrechts Universität, 24098 Kiel, Germany

E. Werner
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106 and Université de Lille 1, UFR de Mathématique, 59655 Villeneuve d’Ascq, France

Keywords: Order statistics, expectations, moments, normal distribution, exponential distribution.
Received by editor(s): March 7, 2005
Received by editor(s) in revised form: June 25, 2005
Published electronically: May 31, 2006
Additional Notes: The first author was partially supported by the Fund for the Promotion of Research at the Technion and by France-Israel Cooperation agreement #3-1350
The first and third authors were partially supported by FP6 Marie Curie Actions, MRTN-CT-2004-511953, PHD
The fourth author was partially supported by an NSF Grant, by a Nato Collaborative Linkage Grant, and by an NSF Advance Opportunity Grant
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society