Bernoulli

  • Bernoulli
  • Volume 24, Number 3 (2018), 2043-2063.

Deviation of polynomials from their expectations and isoperimetry

Lavrentin M. Arutyunyan and Egor D. Kosov

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The article is divided into two parts. In the first part, we study the deviation of a polynomial from its mathematical expectation. This deviation can be estimated from above by Carbery–Wright inequality, so we investigate estimates of the deviation from below. We obtain such type estimates in two different cases: for Gaussian measures and a polynomial of an arbitrary degree and for an arbitrary log-concave measure but only for polynomials of the second degree. In the second part, we deal with the isoperimetric inequality and the Poincaré inequality for probability measures on the real line that are images of the uniform distributions on convex compact sets in $\mathbb{R}^{n}$ under polynomial mappings.

Article information

Source
Bernoulli, Volume 24, Number 3 (2018), 2043-2063.

Dates
Received: April 2016
Revised: November 2016
First available in Project Euclid: 2 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1517540467

Digital Object Identifier
doi:10.3150/16-BEJ919

Mathematical Reviews number (MathSciNet)
MR3757522

Zentralblatt MATH identifier
06839259

Keywords
Carbery–Wright inequality distribution of a polynomial Gaussian measure isoperimetric inequality logarithmically concave measure Poincaré inequality

Citation

Arutyunyan, Lavrentin M.; Kosov, Egor D. Deviation of polynomials from their expectations and isoperimetry. Bernoulli 24 (2018), no. 3, 2043--2063. doi:10.3150/16-BEJ919. https://projecteuclid.org/euclid.bj/1517540467


Export citation

References

  • [1] Arutyunyan, L.M. and Kosov, E.D. (2015). Estimates for integral norms of polynomials on spaces with convex measures. Mat. Sb. 206 3–22.
  • [2] Bobkov, S.G. (1997). Isoperimetric problems in the theory of infinite dimensional probability distributions. Doctoral dissertation, Saint Petersburg.
  • [3] Bobkov, S.G. (1999). Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 1903–1921.
  • [4] Bobkov, S.G. (2000). Remarks on the growth of $L^{p}$-norms of polynomials. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1745 27–35. Berlin: Springer.
  • [5] Bobkov, S.G. (2000). Some generalizations of Prokhorov’s results on Khinchin-type inequalities for polynomials. Theory Probab. Appl. 45 644–647.
  • [6] Bobkov, S.G. (2003). Spectral gap and concentration for some spherically symmetric probability measures. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1807 37–43. Berlin: Springer.
  • [7] Bobkov, S.G. (2009). On the isoperimetric constants for product measures. J. Math. Sci. (N. Y.) 159 47–53.
  • [8] Bobkov, S.G. and Houdré, C. (1997). Isoperimetric constants for product probability measures. Ann. Probab. 25 184–205.
  • [9] Bogachev, V.I. (1998). Gaussian Measures. Mathematical Surveys and Monographs 62. Providence, RI: Amer. Math. Soc.
  • [10] Bogachev, V.I. (2010). Differentiable Measures and the Malliavin Calculus. Mathematical Surveys and Monographs 164. Providence, RI: Amer. Math. Soc.
  • [11] Bogachev, V.I., Kosov, E.D., Nourdin, I. and Poly, G. (2015). Two properties of vectors of quadratic forms in Gaussian random variables. Theory Probab. Appl. 59 208–221.
  • [12] Bogachev, V.I. and Zelenov, G.I. (2015). On convergence in variation of weakly convergent multidimensional distributions. Dokl. Akad. Nauk 461 14–17.
  • [13] Borell, C. (1974). Convex measures on locally convex spaces. Ark. Mat. 12 239–252.
  • [14] Borell, C. (1975). Convex set functions in $d$-space. Period. Math. Hungar. 6 111–136.
  • [15] Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30 207–216.
  • [16] Bourgain, J. (1991). On the distribution of polynomials on high-dimensional convex sets. In Geometric Aspects of Functional Analysis (19891990). Lecture Notes in Math. 1469 127–137. Berlin: Springer.
  • [17] Carbery, A. and Wright, J. (2001). Distributional and $L^{q}$ norm inequalities for polynomials over convex bodies in $\mathbb{R}^{n}$. Math. Res. Lett. 8 233–248.
  • [18] Cheeger, J. (1970). A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis (Papers Dedicated to Salomon Bochner, 1969) 195–199. Princeton, NJ: Princeton Univ. Press.
  • [19] Fradelizi, M. and Guédon, O. (2004). The extreme points of subsets of $s$-concave probabilities and a geometric localization theorem. Discrete Comput. Geom. 31 327–335.
  • [20] Friedland, O. and Sodin, S. (2007). Bounds on the concentration function in terms of the Diophantine approximation. C. R. Math. Acad. Sci. Paris 345 513–518.
  • [21] Gromov, M. and Milman, V.D. (1987). Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces. Compos. Math. 62 263–282.
  • [22] Kannan, R., Lovász, L. and Simonovits, M. (1995). Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 541–559.
  • [23] Klartag, B. (2006). On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16 1274–1290.
  • [24] Littlewood, J.E. and Offord, A.C. (1943). On the number of real roots of a random algebraic equation. III. Rec. Math. [Mat. Sbornik] N.S. 12(54) 277–286.
  • [25] Lovász, L. and Simonovits, M. (1993). Random walks in a convex body and an improved volume algorithm. Random Structures Algorithms 4 359–412.
  • [26] Nazarov, F., Sodin, M. and Volberg, A. (2002). The geometric Kannan–Lovász–Simonovits lemma, dimension-free estimates for the distribution of the values of polynomials, and the distribution of the zeros of random analytic functions. Algebra i Analiz 14 214–234.
  • [27] Nourdin, I., Nualart, D. and Poly, G. (2013). Absolute continuity and convergence of densities for random vectors on Wiener chaos. Electron. J. Probab. 18 no. 22, 19.
  • [28] Nourdin, I. and Poly, G. (2013). Convergence in total variation on Wiener chaos. Stochastic Process. Appl. 123 651–674.
  • [29] Rudelson, M. and Vershynin, R. (2009). Smallest singular value of a random rectangular matrix. Comm. Pure Appl. Math. 62 1707–1739.
  • [30] Rudelson, M. and Vershynin, R. (2015). Small ball probabilities for linear images of high-dimensional distributions. Int. Math. Res. Not. IMRN 19 9594–9617.
  • [31] Sudakov, V.N. and Cirel’son, B.S. (1974). Extremal properties of half-spaces for spherically invariant measures. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 41 14–24, 165.