Minima of sequences of Gaussian random variables

Yehoram Gordon; Alexander Litvak; Carsten Schütt; Elisabeth Werner

doi:10.1016/j.crma.2005.02.003

Probability Theory

Minima of sequences of Gaussian random variables

Presented by: Gilles Pisier

Yehoram Gordon ¹; Alexander Litvak ²; Carsten Schütt ³; Elisabeth Werner ^4,⁵

¹ Technion, Department of Mathematics, Haifa 32000, Israel
² Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
³ Christian Albrechts Universität, Mathematisches Seminar, 24098 Kiel, Germany
⁴ Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA
⁵ Université de Lille 1, UFR de mathématique, 59655 Villeneuve d'Ascq, France

Comptes Rendus. Mathématique, Volume 340 (2005) no. 6, pp. 445-448

Abstract (Original language)
Résumé

For a given sequence of real numbers $a_{1}, \dots, a_{n}$ we denote the k-th smallest one by $k - \min_{1 ⩽ i ⩽ n} a_{i}$ . We show that there exist two absolute positive constants c and C such that for every sequence of positive real numbers $x_{1}, \dots, x_{n}$ and every $k ⩽ n$ one has

c \max_{1 ⩽ j ⩽ k} \frac{k + 1 - j}{\sum_{i = j}^{n} 1 / x_{i}} ⩽ E k - \min_{1 ⩽ i ⩽ n} | x_{i} g_{i} | ⩽ C \ln (k + 1) \max_{1 ⩽ j ⩽ k} \frac{k + 1 - j}{\sum_{i = j}^{n} 1 / x_{i}},

where

g_{i} \in N (0, 1)

,

i = 1, \dots, n

, are independent Gaussian random variables. Moreover, if

k = 1

then the left hand side estimate does not require independence of the

g_{i}

s. Similar estimates hold for

E k - \min_{1 ⩽ i ⩽ n} {| x_{i} g_{i} |}^{p}

as well.

Pour une suite $a_{1}, \dots, a_{n}$ des nombres réels, on note le k-ième plus petit membre par $k - \min_{1 ⩽ i ⩽ n} a_{i}$ . On démontre qu'il existe deux constants positives c et C telles que pour toute suite $x_{1}, \dots, x_{n}$ des nombres réels et pour tout $k ⩽ n$ , on ait

c \max_{1 ⩽ j ⩽ k} \frac{k + 1 - j}{\sum_{i = j}^{n} 1 / x_{i}} ⩽ E k - \min_{1 ⩽ i ⩽ n} | x_{i} g_{i} | ⩽ C \ln (k + 1) \max_{1 ⩽ j ⩽ k} \frac{k + 1 - j}{\sum_{i = j}^{n} 1 / x_{i}} .

Ici

g_{i} \in N (0, 1)

,

i = 1, \dots, n

, sont des variables aléatoires Gaussiennes indépendentes. En plus, si

k = 1

, on n'a pas besoin de l'indépendence des

g_{i}

's pour obtenir l'inégalité du gauche. On démontre également les inégalités correspondantes pour

E k - \min_{1 ⩽ i ⩽ n} {| x_{i} g_{i} |}^{p}

.

Received: 2005-01-13
Accepted: 2005-02-03
Published online: 2005-03-02

DOI: 10.1016/j.crma.2005.02.003

Author's affiliations:

Yehoram Gordon ¹; Alexander Litvak ²; Carsten Schütt ³; Elisabeth Werner ^{4, 5}

¹ Technion, Department of Mathematics, Haifa 32000, Israel
² Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
³ Christian Albrechts Universität, Mathematisches Seminar, 24098 Kiel, Germany
⁴ Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA
⁵ Université de Lille 1, UFR de mathématique, 59655 Villeneuve d'Ascq, France

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     author = {Yehoram Gordon and Alexander Litvak and Carsten Sch\"utt and Elisabeth Werner},
     title = {Minima of sequences of {Gaussian} random variables},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {445--448},
     year = {2005},
     publisher = {Elsevier},
     volume = {340},
     number = {6},
     doi = {10.1016/j.crma.2005.02.003},
     language = {en},
}

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Yehoram Gordon; Alexander Litvak; Carsten Schütt; Elisabeth Werner. Minima of sequences of Gaussian random variables. Comptes Rendus. Mathématique, Volume 340 (2005) no. 6, pp. 445-448. doi: 10.1016/j.crma.2005.02.003

References
Cited by

[1] Y. Gordon; A.E. Litvak; C. Schütt; E. Werner Orlicz norms of sequences of random variables, Ann. Probab., Volume 30 (2002), pp. 1833-1853

[2] S. Mallat, O. Zeitouni, Optimality of the Karhunen–Loeve basis in nonlinear reconstruction, preprint, http://www.ee.technion.ac.il/~zeitouni/openprob.html

[3] O. Zeitouni, private communication

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