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On the minimum of several random variables

Authors:	Y Gordon A E Litvak C Schü tt E Werner

Institution:	Department of Mathematics, Technion, Haifa 32000, Israel ; Department of Mathematics and Statistics Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 ; Mathematisches Seminar, Christian Albrechts Universität, 24098 Kiel, Germany ; Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106 and Université de Lille 1, UFR de Mathématique, 59655 Villeneuve d'Ascq, France

Abstract:	For a given sequence of real numbers $a_{1}, \dots, a_{n}$ , we denote the 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 Gaussian random variables). We show that there exist two absolute positive constants and 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 , 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.

Keywords:	Order statistics expectations moments normal distribution exponential distribution

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