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

DOI:
https://doi.org/10.1090/S0002-9939-06-08453-X

Published electronically:
May 31, 2006

MathSciNet review:
2240681

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a given sequence of real numbers , we denote the th smallest one by . Let 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 and every , one has

**[AB]**B. C. ARNOLD, N. BALAKRISHNAN*Relations, bounds and approximations for order statistics*, Lecture Notes in Statistics, 53, Berlin etc.: Springer-Verlag. viii, 1989. MR**0996887 (90i:62061)****[DN]**H. A. DAVID, H. N. NAGARAJA,*Order statistics*, third ed., Wiley Series in Probability and Statistics. Chichester: John Wiley & Sons, 2003. MR**1994955 (2004f:62007)****[G]**E. D. GLUSKIN,*Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces*, Math. USSR Sbornik, 64 (1989), 85-96. MR**0945901 (89j:46016)****[GLSW1]**Y. GORDON, A. E. LITVAK, C. SCHÜTT, E. WERNER,*Orlicz norms of sequences of random variables*, Ann. of Prob., 30 (2002), 1833-1853. MR**1944007 (2003m:60009)****[GLSW2]**Y. GORDON, A. E. LITVAK, C. SCHÜTT, E. WERNER,*Geometry of spaces between zonoids and polytopes*, Bull. Sci. Math., 126 (2002), 733-762. MR**1941083 (2003i:52006)****[GLSW3]**Y. GORDON, A. E. LITVAK, C. SCHÜTT, E. WERNER,*Minima of sequences of Gaussian random variables*, C. R. Acad. Sci. Paris, Sér. I Math., 340 (2005), 445-448. MR**2135327****[HLP]**G. H. HARDY, J. E. LITTLEWOOD AND G. POLYA,*Inequalities*, second ed., Cambridge, The University Press. XII, 1952. MR**0046395 (13:727e)****[KS1]**S. KWAPIEN, C. SCHÜTT,*Some combinatorial and probabilistic inequalities and their application to Banach space theory*, Studia Math. 82 (1985), 91-106. MR**0809774 (87h:46042)****[KS2]**S. KWAPIEN, C. SCHÜTT,*Some combinatorial and probabilistic inequalities and their application to Banach space theory. II*, Studia Math. 95 (1989), 141-154. MR**1038501 (91e:46027)****[S]**Z. SID´AK,*Rectangular confidence regions for the means of multivariate normal distributions*, J. Am. Stat. Assoc. 62 (1967), 626-633. MR**0216666 (35:7495)**

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Additional 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

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

Email:
emw2@po.cwru.edu

DOI:
https://doi.org/10.1090/S0002-9939-06-08453-X

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