## Electronic Communications in Probability

### Uniform estimates for averages of order statistics of matrices

#### Abstract

We prove uniform estimates for the expected value of averages of order statistics of matrices interms of their largest entries. As an application, we obtain similar probabilistic estimates for $\ell_p$ norms via real interpolation.

#### Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 27, 12 pp.

Dates
Accepted: 19 March 2015
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465320954

Digital Object Identifier
doi:10.1214/ECP.v20-3992

Mathematical Reviews number (MathSciNet)
MR3327866

Zentralblatt MATH identifier
1318.62153

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 05A20: Combinatorial inequalities

Rights

#### Citation

Lechner, Richard; Passenbrunner, Markus; Prochno, Joscha. Uniform estimates for averages of order statistics of matrices. Electron. Commun. Probab. 20 (2015), paper no. 27, 12 pp. doi:10.1214/ECP.v20-3992. https://projecteuclid.org/euclid.ecp/1465320954

#### References

• Gluskin, E. D. Estimates of the norms of certain $p$-absolutely summing operators. (Russian) Funktsional. Anal. i Prilozhen. 12 (1978), no. 2, 24–31, 95.
• Gordon, Yehoram; Litvak, Alexander; SchÃ¼tt, Carsten; Werner, Elisabeth. Orlicz norms of sequences of random variables. Ann. Probab. 30 (2002), no. 4, 1833–1853.
• Gordon, Yehoram; Litvak, Alexander; SchÃ¼tt, Carsten; Werner, Elisabeth. Geometry of spaces between polytopes and related zonotopes. Bull. Sci. Math. 126 (2002), no. 9, 733–762.
• Gordon, Yehoram; Litvak, Alexander; SchÃ¼tt, Carsten; Werner, Elisabeth. Minima of sequences of Gaussian random variables. C. R. Math. Acad. Sci. Paris 340 (2005), no. 6, 445–448.
• Gordon, Y.; Litvak, A. E.; SchÃ¼tt, C.; Werner, E. On the minimum of several random variables. Proc. Amer. Math. Soc. 134 (2006), no. 12, 3665–3675 (electronic).
• Gordon, Yehoram; Litvak, Alexander E.; SchÃ¼tt, Carsten; Werner, Elisabeth. Uniform estimates for order statistics and Orlicz functions. Positivity 16 (2012), no. 1, 1–28.
• Johnson, W. B.; Maurey, B.; Schechtman, G.; Tzafriri, L. Symmetric structures in Banach spaces. Mem. Amer. Math. Soc. 19 (1979), no. 217, v+298 pp.
• KwapieÅ„, Stanisław; SchÃ¼tt, Carsten. Some combinatorial and probabilistic inequalities and their application to Banach space theory. Studia Math. 82 (1985), no. 1, 91–106.
• KwapieÅ„, Stanisław; SchÃ¼tt, Carsten. Some combinatorial and probabilistic inequalities and their application to Banach space theory. II. Studia Math. 95 (1989), no. 2, 141–154.
• Lindenstrauss, Joram; Tzafriri, Lior. Classical Banach spaces. I. Sequence spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. Springer-Verlag, Berlin-New York, 1977. xiii+188 pp. ISBN: 3-540-08072-4
• Montgomery-Smith, Stephen; Semenov, Evgueni. Random rearrangements and operators. Voronezh Winter Mathematical Schools, 157–183, Amer. Math. Soc. Transl. Ser. 2, 184, Amer. Math. Soc., Providence, RI, 1998.
• Prochno, Joscha. A combinatorial approach to Musielak-Orlicz spaces. Banach J. Math. Anal. 7 (2013), no. 1, 132–141.
• Prochno, Joscha; SchÃ¼tt, Carsten. Combinatorial inequalities and subspaces of $L_ 1$. Studia Math. 211 (2012), no. 1, 21–39.
• Rao, M. M.; Ren, Z. D. Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, 146. Marcel Dekker, Inc., New York, 1991. xii+449 pp. ISBN: 0-8247-8478-2
• SchÃ¼tt, Carsten. On the positive projection constant. Studia Math. 78 (1984), no. 2, 185–198.
• SchÃ¼tt, Carsten. Lorentz spaces that are isomorphic to subspaces of $L^ 1$. Trans. Amer. Math. Soc. 314 (1989), no. 2, 583–595.
• SchÃ¼tt, Carsten. On the embedding of $2$-concave Orlicz spaces into $L^ 1$. Studia Math. 113 (1995), no. 1, 73–80.
• Wisła, Marek. Extreme points and stable unit balls in Orlicz sequence spaces. Arch. Math. (Basel) 56 (1991), no. 5, 482–490.