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Tytuł artykułu
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Języki publikacji
Abstrakty
We prove logarithmic Sobolev inequalities and concentration results for convex functions and a class of product random vectors. The results are used to derive tail and moment inequalities for chaos variables (in the spirit of Talagrand and Arcones-Gine). We also show that the same proof may be used for chaoses generated by log-concave random variables, recovering results by Łochowski, and present an application to exponential integrability of Rademacher chaos.
Wydawca
Rocznik
Tom
Strony
221--238
Opis fizyczny
Bibliogr. 19 poz.
Twórcy
autor
- Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P. O. Box 21, 00-956 Warszawa 10, Poland, R.Adamczak@impan.gov.pl
Bibliografia
- [1] M. Arcones and E. Giné, On decoupling, series expansion and tail behaviour of chaos processes, J. Theoret. Probab. 6 (1993), 101-122.
- [2] S. G. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal. 163 (1999), 1-28.
- [3] C. Borell, On a Taylor series of a Wiener polynomial, in: Seminar Notes on Multiple Stochastic Integration, Polynomial Chaos and Their Integration, Case Western Reserve Univ., Cleveland, 1984.
- [4] S. Boucheron, О. Bousquet, G. Lugosi and R Massart, Moment inequalities for functions of independent random variables, Ann. Probab. 33 (2005), 514-560.
- [5] P. Hitczenko, S. Kwapień, W. V. Li, G. Schechtman, T. Schlumprecht and J. Zinn, Hypercontractivity and comparison of moments of iterated maxima and minima of independent random variables, Electronic J. Probab. 3 (1998).
- [6] R. Latała, Tail and moment estimates for sums of independent random vectors with logarithmically concave tails, Studia Math. 118 (1996), 301-304.
- [7] —, Tail and moment estimates for some types of chaos, Studia Math. 135 (1999), 39-53.
- [8] —, Estimation of moments and tails of Gaussian chaoses, preprint, 2005, http://www.arxiv.org/abs/math.PR/0505313.
- [9] R. Latała and R. Łochowski, Moment and tail estimates for multidimensional chaos generated by positive random variables with logarithmically concave tails, Progr. Probab. 56 (2003), 77-92.
- [10] R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, in: Geometric Aspects of Functional Analysis, Israel Seminar (GAFA) 1996-2000, Lecture Notes in Math. 1745, Springer, Berlin, 2000, 147-168.
- [11] M. Ledoux, On Talagrand’s deviation inequalities for product measures, ESAIM: Probab. Statist. 1 (1996), 63-87.
- [12] —, The Concentration of Measure Phenomenon, Math. Surveys Monogr. 89, Amer. Math. Soc., 2001.
- [13] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, New York, 1991.
- [14] R. Łochowski, Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails, preprint.
- [15] V. H. de la Peña and E. Giné, Decoupling. From Dependence to Independence, Springer, 1999.
- [16] V. H. de la Peña and S. Montgomery-Smith, Bounds on the tail probability of U- statistics and quadratic forms, Bull. Amer. Math. Soc. 31 (1994), 223-227.
- [17] P. M. Samson, Concentration of measure inequalities for Markov chains and Ф- mixing processes, Ann. Probab. 28 (2000), 416-461.
- [18] M. Talagrand, New concentration inequalities in product spaces, Invent. Math. 126 (1996), 505-563.
- [19] —, A new look at independence, Ann. Probab. 24 (1996), 1-34.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BAT5-0006-0054