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Gaussian mixtures: entropy and geometric inequalities. (English) Zbl 1428.60036
Summary: A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to $$e^{-|t|^{p}}$$ and symmetric $$p$$-stable random variables, where $$p\in(0,2]$$. We obtain various sharp moment and entropy comparison estimates for weighted sums of independent Gaussian mixtures and investigate extensions of the B-inequality and the Gaussian correlation inequality in the context of Gaussian mixtures. We also obtain a correlation inequality for symmetric geodesically convex sets in the unit sphere equipped with the normalized surface area measure. We then apply these results to derive sharp constants in Khinchine inequalities for vectors uniformly distributed on the unit balls with respect to $$p$$-norms and provide short proofs to new and old comparison estimates for geometric parameters of sections and projections of such balls.

##### MSC:
 60E15 Inequalities; stochastic orderings 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 52A40 Inequalities and extremum problems involving convexity in convex geometry 94A17 Measures of information, entropy
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##### References:
 [1] Arora, S. and Kannan, R. (2001). Learning mixtures of arbitrary Gaussians. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing 247–257. ACM, New York. · Zbl 1323.68440 [2] Artstein, S., Ball, K. M., Barthe, F. and Naor, A. (2004). Solution of Shannon’s problem on the monotonicity of entropy. J. Amer. Math. Soc.17 975–982. · Zbl 1062.94006 [3] Artstein-Avidan, S., Giannopoulos, A. and Milman, V. D. (2015). Asymptotic Geometric Analysis, Part I. Mathematical Surveys and Monographs202. Amer. Math. Soc., Providence, RI. · Zbl 1337.52001 [4] Averkamp, R. and Houdré, C. (2003). Wavelet thresholding for non-necessarily Gaussian noise: Idealism. Ann. Statist.31 110–151. · Zbl 1102.62329 [5] Baernstein, A. II and Culverhouse, R. C. (2002). Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions. Studia Math.152 231–248. · Zbl 0997.60009 [6] Ball, K. (1986). Cube slicing in $$\textbf{R}^{n}$$. Proc. Amer. Math. Soc.97 465–473. · Zbl 0601.52005 [7] Ball, K., Nayar, P. and Tkocz, T. (2016). A reverse entropy power inequality for log-concave random vectors. Studia Math.235 17–30. · Zbl 1407.94055 [8] Barthe, F. (1995). Mesures unimodales et sections des boules $$B^{n}_{p}$$. C. R. Acad. Sci. Paris Sér. I Math.321 865–868. · Zbl 0876.46011 [9] Barthe, F., Guédon, O., Mendelson, S. and Naor, A. (2005). A probabilistic approach to the geometry of the $$l^{n}_{p}$$-ball. Ann. Probab.33 480–513. · Zbl 1071.60010 [10] Barthe, F. and Naor, A. (2002). Hyperplane projections of the unit ball of $$l^{n}_{p}$$. Discrete Comput. Geom.27 215–226. · Zbl 0999.52003 [11] Bobkov, S. G. and Chistyakov, G. P. (2015). Entropy power inequality for the Rényi entropy. IEEE Trans. Inform. Theory61 708–714. · Zbl 1359.94300 [12] Bobkov, S. G. and Houdré, C. (1996). Characterization of Gaussian measures in terms of the isoperimetric property of half-spaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 228 31–38, 356. [13] Bobkov, S. G. and Houdré, C. (1997). Isoperimetric constants for product probability measures. Ann. Probab.25 184–205. · Zbl 0878.60013 [14] Böröczky, K. J., Lutwak, E., Yang, D. and Zhang, G. (2012). The log-Brunn–Minkowski inequality. Adv. Math.231 1974–1997. · Zbl 1258.52005 [15] Cordero-Erausquin, D., Fradelizi, M. and Maurey, B. (2004). The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. J. Funct. Anal.214 410–427. · Zbl 1073.60042 [16] Dasgupta, S. (1999). Learning mixtures of Gaussians. In 40th Annual Symposium on Foundations of Computer Science (New York, 1999) 634–644. IEEE Computer Soc., Los Alamitos, CA. [17] Eskenazis, A. and Nayar, P. and Tkocz, T. (2018). Sharp comparison of moments and the log-concave moment problem. Preprint. Available at arXiv:1801.07597.. · Zbl 1435.60019 [18] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II. 2nd ed. Wiley, New York. · Zbl 0219.60003 [19] Gozlan, N. and Léonard, C. (2010). Transport inequalities. A survey. Markov Process. Related Fields16 635–736. · Zbl 1229.26029 [20] Guédon, O., Nayar, P. and Tkocz, T. (2014). Concentration inequalities and geometry of convex bodies. In Analytical and Probabilistic Methods in the Geometry of Convex Bodies. IMPAN Lect. Notes2 9–86. Polish Acad. Sci. Inst. Math., Warsaw. · Zbl 1320.52007 [21] Haagerup, U. (1981). The best constants in the Khintchine inequality. Studia Math.70 231–283 (1982). · Zbl 0501.46015 [22] Hardy, G. H., Littlewood, J. E. and Pólya, G. (1988). Inequalities. Cambridge Univ. Press, Cambridge. Reprint of the 1952 edition. [23] Klartag, B. and Vershynin, R. (2007). Small ball probability and Dvoretzky’s theorem. Israel J. Math.157 193–207. · Zbl 1120.46003 [24] Koldobsky, A. (1998). An application of the Fourier transform to sections of star bodies. Israel J. Math.106 157–164. · Zbl 0916.52002 [25] Koldobsky, A. (2005). Fourier Analysis in Convex Geometry. Mathematical Surveys and Monographs116. Amer. Math. Soc., Providence, RI. · Zbl 1082.52002 [26] Koldobsky, A. and Zymonopoulou, M. (2003). Extremal sections of complex $$l_{p}$$-balls, \(0
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