On the minimum of several random variables
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- by Y. Gordon, A. E. Litvak, C. Schütt and E. Werner
- Proc. Amer. Math. Soc. 134 (2006), 3665-3675
- DOI: https://doi.org/10.1090/S0002-9939-06-08453-X
- Published electronically: May 31, 2006
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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 \[ c \max _{1 \leq j \leq k}\ \frac {k+1-j}{\sum _{i=j}^n 1/x_i } \leq \mathbb E k\mbox {-}\min _{1\leq i\leq n} |x_{i} \xi _{i}| \leq C \ln (k+1) \max _{1 \leq j \leq k}\ \frac {k+1-j}{\sum _{i=j}^n 1/x_i}, \] 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 ${k\mbox {-}\min }_{1\leq i\leq n} |x_{i} \xi _{i}|$ as well.References
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Bibliographic Information
- Y. Gordon
- Affiliation: Department of Mathematics, Technion, Haifa 32000, Israel
- Email: gordon@techunix.technion.ac.il
- A. E. Litvak
- Affiliation: Department of Mathematics and Statistics Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
- MR Author ID: 367520
- Email: alexandr@math.ualberta.ca
- C. Schütt
- Affiliation: Mathematisches Seminar, Christian Albrechts Universität, 24098 Kiel, Germany
- Email: schuett@math.uni-kiel.de
- 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
- MR Author ID: 252029
- ORCID: 0000-0001-9602-2172
- Email: emw2@po.cwru.edu
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 3665-3675
- MSC (2000): Primary 62G30, 60E15, 60G51
- DOI: https://doi.org/10.1090/S0002-9939-06-08453-X
- MathSciNet review: 2240681