Gaussian mixtures: entropy and geometric inequalities. (English) Zbl 1428.60036

Eskenazis, Alexandros; Nayar, Piotr; Tkocz, Tomasz

Ann. Probab. 46, No. 5, 2908-2945 (2018).

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

Keywords:

Gaussian measure; Gaussian mixture; Khinchine inequality; entropy; B-inequality; small ball probability; correlation inequalities; extremal sections and projections of \(\ell_{p}\)-balls

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 · doi:10.1090/S0894-0347-04-00459-X
[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 · doi:10.1214/aos/1046294459
[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 · doi:10.4064/sm152-3-3
[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 · doi:10.1007/s00454-001-0066-3
[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 · doi:10.1109/TIT.2014.2383379
[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 · doi:10.1214/aop/1024404284
[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 · doi:10.1016/j.aim.2012.07.015
[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 · doi:10.1016/j.jfa.2003.12.001
[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 · doi:10.1016/j.aim.2018.06.014
[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 · doi:10.4064/sm-70-3-231-283
[22]	Hardy, G. H., Littlewood, J. E. and Pólya, G. (1988). Inequalities. Cambridge Univ. Press, Cambridge. Reprint of the 1952 edition. · Zbl 0634.26008
[23]	Klartag, B. and Vershynin, R. (2007). Small ball probability and Dvoretzky’s theorem. Israel J. Math.157 193–207. · Zbl 1120.46003 · doi:10.1007/s11856-006-0007-1
[24]	Koldobsky, A. (1998). An application of the Fourier transform to sections of star bodies. Israel J. Math.106 157–164. · Zbl 0916.52002 · doi:10.1007/BF02773465
[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, \(0<p\le2\). Studia Math.159 185–194. · Zbl 1053.52005 · doi:10.4064/sm159-2-2
[27]	Koldobsky, A. L. and Montgomery-Smith, S. J. (1996). Inequalities of correlation type for symmetric stable random vectors. Statist. Probab. Lett.28 91–97. · Zbl 0855.60016 · doi:10.1016/0167-7152(95)00096-8
[28]	König, H. (2014). On the best constants in the Khintchine inequality for Steinhaus variables. Israel J. Math.203 23–57. · Zbl 1314.46017
[29]	Latała, R. (2002). On some inequalities for Gaussian measures. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) 813–822. Higher Ed. Press, Beijing. · Zbl 1015.60011
[30]	Latała, R. and Matlak, D. (2017). Royen’s proof of the Gaussian correlation inequality. In Geometric Aspects of Functional Analysis. Lecture Notes in Math.2169 265–275. Springer, Cham. · Zbl 1366.60058
[31]	Latała, R. and Oleszkiewicz, K. (1995). A note on sums of independent uniformly distributed random variables. Colloq. Math.68 197–206. · Zbl 0821.60027 · doi:10.4064/cm-68-2-197-206
[32]	Latała, R. and Oleszkiewicz, K. (2005). Small ball probability estimates in terms of widths. Studia Math.169 305–314. · Zbl 1073.60043 · doi:10.4064/sm169-3-6
[33]	Lewis, T. M. and Pritchard, G. (1999). Correlation measures. Electron. Commun. Probab.4 77–85. · Zbl 0936.60013 · doi:10.1214/ECP.v4-1008
[34]	Lewis, T. M. and Pritchard, G. (2003). Tail properties of correlation measures. J. Theoret. Probab.16 771–788. · Zbl 1030.60013 · doi:10.1023/A:1025680718292
[35]	Lieb, E. H. (1978). Proof of an entropy conjecture of Wehrl. Comm. Math. Phys.62 35–41. · Zbl 0385.60089 · doi:10.1007/BF01940328
[36]	Livne Bar-on, A. (2014). The (B) conjecture for uniform measures in the plane. In Geometric Aspects of Functional Analysis. Lecture Notes in Math.2116 341–353. Springer, Cham. · Zbl 1333.52002
[37]	Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Mathematics in Science and Engineering143. Academic Press, New York. · Zbl 0437.26007
[38]	Marshall, A. W. and Proschan, F. (1965). An inequality for convex functions involving majorization. J. Math. Anal. Appl.12 87–90. · Zbl 0145.28601 · doi:10.1016/0022-247X(65)90056-9
[39]	Marsiglietti, A. (2016). On the improvement of concavity of convex measures. Proc. Amer. Math. Soc.144 775–786. · Zbl 1334.28007 · doi:10.1090/proc/12694
[40]	Memarian, Y. (2015). On a correlation inequality for Cauchy type measures. New Zealand J. Math.45 53–64. · Zbl 1333.52004
[41]	Meyer, M. and Pajor, A. (1988). Sections of the unit ball of \(l^{n}_{p}\). J. Funct. Anal.80 109–123. · Zbl 0667.46004 · doi:10.1016/0022-1236(88)90068-7
[42]	Nayar, P. and Oleszkiewicz, K. (2012). Khinchine type inequalities with optimal constants via ultra log-concavity. Positivity16 359–371. · Zbl 1260.60031 · doi:10.1007/s11117-011-0130-z
[43]	Paouris, G. and Valettas, P. (2018). A small deviation inequality for convex functions. Ann. Probab. 46 1441–1454. · Zbl 1429.60022 · doi:10.1214/17-AOP1206
[44]	Paouris, G. and Valettas, P. (2016). Variance estimates and almost Euclidean structure. Preprint. Available at arXiv:1703.10244. · Zbl 1404.46014 · doi:10.1016/j.aim.2018.05.022
[45]	Rachev, S. T. and Rüschendorf, L. (1991). Approximate independence of distributions on spheres and their stability properties. Ann. Probab.19 1311–1337. · Zbl 0732.62014 · doi:10.1214/aop/1176990346
[46]	Royen, T. (2014). A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions. Far East J. Theor. Stat.48 139–145. · Zbl 1314.60070
[47]	Saroglou, C. (2015). Remarks on the conjectured log-Brunn–Minkowski inequality. Geom. Dedicata177 353–365. · Zbl 1326.52010 · doi:10.1007/s10711-014-9993-z
[48]	Saroglou, C. (2016). More on logarithmic sums of convex bodies. Mathematika62 818–841. · Zbl 1352.52001 · doi:10.1112/S0025579316000061
[49]	Schechtman, G. and Zinn, J. (1990). On the volume of the intersection of two \(L^{n}_{p}\) balls. Proc. Amer. Math. Soc.110 217–224. · Zbl 0704.60017
[50]	Shannon, C. E. and Weaver, W. (1949). The Mathematical Theory of Communication. Univ. Illinois Press, Urbana, IL. · Zbl 0041.25804
[51]	Simon, T. (2011). Multiplicative strong unimodality for positive stable laws. Proc. Amer. Math. Soc.139 2587–2595. · Zbl 1222.60020 · doi:10.1090/S0002-9939-2010-10697-4
[52]	Stam, A. J. (1959). Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control2 101–112. · Zbl 0085.34701 · doi:10.1016/S0019-9958(59)90348-1
[53]	Szarek, S. J. (1976). On the best constants in the Khinchin inequality. Studia Math.58 197–208. · Zbl 0424.42014 · doi:10.4064/sm-58-2-197-208
[54]	Vershynin, R. (2009). Lectures in geometric functional analysis. Lecture notes, available at https://www.math.uci.edu/ rvershyn/papers/GFA-book.pdf.
[55]	Weron, A. (1984). Stable processes and measures: A survey. In Probability Theory on Vector Spaces, III (Lublin, 1983). Lecture Notes in Math.1080 306–364. Springer, Berlin. · Zbl 0548.60005
[56]	Whittle, P. (1960). Bounds for the moments of linear and quadratic forms in independent variables. Teor. Verojatnost. i Primenen.5 331–335. · Zbl 0101.12003
[57]	Yu, Y. (2008). Letter to the editor: On an inequality of Karlin and Rinott concerning weighted sums of i.i.d. random variables. Adv. in Appl. Probab.40 1223–1226. · Zbl 1369.60007 · doi:10.1239/aap/1231340171