Borell and Landau–Shepp Inequalities for Cauchy-Type Measures

• Autorzy: Byczkowski Tomasz , Żak Tomasz

Identyfikatory

DOI

10.37190/0208-4147.41.1.9

• Języki publikacji: EN

Abstrakty

EN

We investigate various inequalities for the one-dimensional Cauchy measure. We also consider analogous properties for onedimensional sections of multidimensional isotropic Cauchy measures. The paper is a continuation of our previous investigations [2], where we found, among intervals with fixed measure, the ones with the extremal measure of the boundary. Here for the above mentioned measures we investigate inequalities that are analogous to those proved for Gaussian measures by Borell [1] and by Landau and Shepp [5]. We also consider a 1-symmetrization for Cauchy measures, analogous to the one defined for Gaussian measures by Ehrhard [3], and we prove the concavity of this operation.

Słowa kluczowe

EN

Cauchy distribution; Borell inequality; Landau–Shepp inequality; Ehrhard symmetrization

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2021
• Tom: Vol. 41, Fasc. 1
• Strony: 129--152
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Byczkowski Tomasz

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland
• Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

autor

Żak Tomasz

• Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Bibliografia

• [1] C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), 207-216.
• [2] T. Byczkowski and T. Żak, Extremal properties of one-dimensional Cauchy-type measures, Probab. Math. Statist. 35 (2015), 247-266.
• [3] A. Ehrhard, Symétrisation dans l’espace de Gauss, Math. Scand. 53 (1983), 281-301.
• [4] S. Kwapień and J. Sawa, On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets, Studia Math. 105 (1993), 173-187.
• [5] H. J. Landau and L. A. Shepp, On the supremum of a Gaussian process, Sankhyā 32 (1970), 369-378.
• [6] R. Latała and K. Oleszkiewicz, Gaussian measures of dilations of convex symmetric sets, Ann. Probab. 27 (1999), 1922-1938.
• [7] E. Milman and L. Rotem, Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures, Adv. Math. 262 (2014), 867-908.
• [8] A. Prekopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335-343.
• [9] V. N. Sudakov and B. S. Tsirel’son, Extremal properties of half-spaces for spherically invariant measures, Zap. Nauchn. Sem. LOMI 41 (1974), 14-24 (in Russian).

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).