## The Annals of Probability

### A Gaussian small deviation inequality for convex functions

#### Abstract

Let $Z$ be an $n$-dimensional Gaussian vector and let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a convex function. We prove that

$\mathbb{P}(f(Z)\leq \mathbb{E}f(Z)-t\sqrt{\operatorname{Var}f(Z)})\leq\exp (-ct^{2}),$ for all $t>1$ where $c>0$ is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.

#### Article information

Source
Ann. Probab., Volume 46, Number 3 (2018), 1441-1454.

Dates
Revised: May 2017
First available in Project Euclid: 12 April 2018

https://projecteuclid.org/euclid.aop/1523520021

Digital Object Identifier
doi:10.1214/17-AOP1206

Mathematical Reviews number (MathSciNet)
MR3785592

Zentralblatt MATH identifier
06894778

#### Citation

Paouris, Grigoris; Valettas, Petros. A Gaussian small deviation inequality for convex functions. Ann. Probab. 46 (2018), no. 3, 1441--1454. doi:10.1214/17-AOP1206. https://projecteuclid.org/euclid.aop/1523520021

#### References

• [1] Bogachev, V. I. (1998). Gaussian Measures. Mathematical Surveys and Monographs 62. Amer. Math. Soc., Providence, RI.
• [2] Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30 207–216.
• [3] Borell, C. (2003). The Ehrhard inequality. C. R. Math. Acad. Sci. Paris 337 663–666.
• [4] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Non-asymptotic Theory of Independence. Oxford Univ. Press, Oxford.
• [5] Chatterjee, S. (2014). Superconcentration and Related Topics. Springer, Cham.
• [6] Chen, L. H. Y. (1982). An inequality for the multivariate normal distribution. J. Multivariate Anal. 12 306–315.
• [7] 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.
• [8] Ehrhard, A. (1983). Symétrisation dans l’espace de Gauss. Math. Scand. 53 281–301.
• [9] Eskenazis, A., Nayar, P. and Tkocz, T. (2016). Gaussian mixtures: Entropy and geometric inequalities. Preprint. Available at https://arxiv.org/abs/1611.04921.
• [10] Grafakos, L. (2004). Classical and Modern Fourier Analysis. Pearson Education, Upper Saddle River, NJ.
• [11] Indyk, P. (2001). Algorithmic applications of low-distortion geometric embeddings. In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001) 10–33. IEEE Computer Soc., Los Alamitos, CA.
• [12] Ivanisvili, P. and Volberg, A. (2015). Bellman partial differential equation and the hill property for classical isoperimetric problems. Preprint. Available at https://arxiv.org/abs/1506.03409.
• [13] Johnson, W. B. and Lindenstrauss, J. (1984). Extensions of Lipschitz mappings into a Hilbert space. In Conference in Modern Analysis and Probability (New Haven, Conn., 1982). 189–206. Amer. Math. Soc., Providence, RI.
• [14] Johnson, W. B. and Naor, A. (2010). The Johnson–Lindenstrauss lemma almost characterizes Hilbert space, but not quite. Discrete Comput. Geom. 43 542–553.
• [15] Klartag, B. and Vershynin, R. (2007). Small ball probability and Dvoretzky’s theorem. Israel J. Math. 157 193–207.
• [16] Kushilevitz, E., Ostrovsky, R. and Rabani, Y. (2000). Efficient search for approximate nearest neighbor in high dimensional spaces. SIAM J. Comput. 30 457–474.
• [17] Kwapień, S. (1994). A remark on the median and the expectation of convex functions of Gaussian vectors. In Probability in Banach Spaces, 9 (Sandjberg, 1993). Progress in Probability 35 271–272. Birkhäuser, Boston, MA.
• [18] Latała, R. (1996). A note on the Ehrhard inequality. Studia Math. 118 169–174.
• [19] Latała, R. and Oleszkiewicz, K. (2005). Small ball probability estimates in terms of widths. Studia Math. 169 305–314.
• [20] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
• [21] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin.
• [22] Litvak, A. E., Milman, V. D. and Schechtman, G. (1998). Averages of norms and quasi-norms. Math. Ann. 312 95–124.
• [23] Milman, V. D. (1971). New proof of the theorem of A. Dvoretzky on sections of convex bodies (in Russian). Funkcional. Anal. i Prilozen. 5 28–37.
• [24] Milman, V. D. and Schechtman, G. (1986). Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Math. 1200. Springer, Berlin.
• [25] Nayar, P. and Tkocz, T. (2013). A note on a Brunn-Minkowski inequality for the Gaussian measure. Proc. Amer. Math. Soc. 141 4027–4030.
• [26] Neeman, J. and Paouris, G. (2016). An interpolation proof of Ehrhard’s inequality. Preprint. Available at https://arxiv.org/abs/1605.07233.
• [27] Paouris, G., Pivovarov, P. and Valettas, P. (2017). On a quantitative reversal of Alexandrov’s inequality. Trans. Amer. Math. Soc. To appear. Available at https://arxiv.org/abs/1702.05762.
• [28] Paouris, G. and Valettas, P. (2015). On Dvoretzky’s theorem for subspaces of $L_{p}$. Preprint. Available at http://arxiv.org/abs/1510.07289.
• [29] Paouris, G. and Valettas, P. (2017). Variance estimates and almost Euclidean structure. Preprint. Available at https://arxiv.org/abs/1703.10244.
• [30] Paouris, G., Valettas, P. and Zinn, J. (2017). Random version of Dvoretzky’s theorem in $\ell_{p}^{n}$. Stochastic Process. Appl. 127 3187–3227.
• [31] Schechtman, G. (2006). Two observations regarding embedding subsets of Euclidean spaces in normed spaces. Adv. Math. 200 125–135.
• [32] Schechtman, G. (2007). The random version of Dvoretzky’s theorem in $\ell^{n}_{\infty}$. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1910 265–270. Springer, Berlin.
• [33] Sudakov, V. N. and Tsirel’son, B. S. (1974). Extremal properties of half-spaces for spherically invariant measures (in Russian). Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 14–24.
• [34] van Handel, R. (2017). The Borell-Ehrhard game. Probab. Theory Related Fields To appear. Available at DOI:10.1007/s00440-017-0762-4.
• [35] Vempala, S. S. (2004). The Random Projection Method. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 65. Amer. Math. Soc., Providence, RI.