## Electronic Journal of Probability

### Hypercontractivity and Comparison of Moments of Iterated Maxima and Minima of Independent Random Variables

#### Abstract

We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.'s. It turns out that the idea of hypercontractivity for minima is closely related to small ball probabilities and Gaussian correlation inequalities.

#### Article information

Source
Electron. J. Probab., Volume 3 (1998), paper no. 2, 26 pp.

Dates
Accepted: 7 January 1998
First available in Project Euclid: 29 January 2016

https://projecteuclid.org/euclid.ejp/1454101762

Digital Object Identifier
doi:10.1214/EJP.v3-24

Mathematical Reviews number (MathSciNet)
MR1491527

Zentralblatt MATH identifier
0889.60003

Rights

#### Citation

Hitczenko, Pawel; Kwapien, Stanislaw; Li, Wenbo; Schechtman, Gideon; Schlumprecht, Thomas; Zinn, Joel. Hypercontractivity and Comparison of Moments of Iterated Maxima and Minima of Independent Random Variables. Electron. J. Probab. 3 (1998), paper no. 2, 26 pp. doi:10.1214/EJP.v3-24. https://projecteuclid.org/euclid.ejp/1454101762

#### References

• Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6, 170–176.
• Asmar, N. H., Montgomery-Smith, S.J. (1994). On the distribution of Sidon series. Ark. Mat. 31, 13–26.
• Beckner, W. (1975). Inequalities in Fourier Analysis. Ann. of Math. 102, 159–182.
• Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular variation. Cambridge University Press, 491 pp.
• Bonami, A. (1970). Etude des coefficients de Fourier des fonctiones de $L_p(G)$. Ann. de l'Institut Fourier 20, 335–402.
• Borell, C. (1979). On the integrability of Banach space valued Walsh polynomials. Séminaire de Probabilité XIII, Lecture in Math. 721, 1–3.
• Borell, C. (1984). On polynomial chaos and itegrability. Probability and Mathematical Statistics 3, 191–203.
• de la Peńa, V., Montgomery-Smith, S.J. and Szulga, J. (1994). Contraction and decoupling inequalities for multilinear forms and U-statistics. Ann. of Prob. 22, 1745–1765.
• Gordon, Y. (1987). Elliptically contoured distributions. Prob. Theory Rel. Fields 76, 429–438.
• Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math 97, 1061–1083.
• Khatri, C.G. (1967). On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Statist. 38, 1853–1867.
• Krakowiak, W. and Szulga, J. (1988). Hypercontraction principle and random multilinear forms. Probability Theory and Related Fields 77, 325–342.
• Kuelbs, J., Li, W.V. and Shao, Q. (1995). Small ball estimates for fractional Brownian motion under H"older norm and Chung's functional LIL. J. Theor. Prob. 8, 361-386.
• Ledoux, M. and Talagrand, M. (1991). Probability on Banach Spaces, Springer, Berlin.
• Lewandowski, M., Ryznar, M. and Żak, T. (1992). Stable measure of a small ball, Proc. Amer. Math. Soc., 115, 489–494.
• Li, W. V. and Shao, Q. (1996). A note on the existence of the small ball constant for sup-norm for fractional Brownian motions assuming the correlation conjecture. Preprint.
• Marcus, M. B. and Shepp, L. (1972). Sample behavior of Gaussian processes. Proc. of the Sixth Berkeley Symposium on Math. Statist. and Prob. vol. 2, 423–421.
• Nelson, E. (1966). A quartic interaction in two dimensions, in Mathematical Theory of Elementary Particles, M.I.T. Press, 69–73.
• Pitt, L.D. (1977). A Gaussian correlation inequality for symmetric convex sets. Ann. of Prob. 5, 470–474.
• Rychlik, E. (1993). Some necessary and sufficient conditions of $(p,q)$ hypercontractivity. Preprint.
• Schechtman, G., Schlumprecht, T. and Zinn, J. (1995). On the Gaussian measure of the intersection of symmetric, convex sets. Preprint.
• Sǐdák, Z. (1967) Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc. 62, 626–633.
• Sǐdák, Z. (1968). On multivariate normal probabilities of rectangles: their dependence on correlations. Ann. Math. Statist. 39, 1425–1434.
• Slepian, D. (1962). The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463–501.
• Szarek, S. (1991). Conditional numbers of random matrices. J. Complexity, 7, 131–149.
• Szarek, S. and Werner, E. (1995). Personal Communication.
• Szulga, J. (1990). A note on hypercontractivity of stable random variables. Ann. of Prob. 18, 1746–1758.
• Talagrand, M. (1994). The small ball problem for the Brownian sheet. Ann. of Probab. 22, 1331–1354.
• Tong, Y. L. (1980). Probability Inequalities in Multivariate Distributions, Academic Press, New York.