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Contents of PMS, Vol. 42, Fasc. 1,
pages 163 - 175
DOI: 10.37190/0208-4147.00036
Published online 8.8.2022
 

Ehrhard-type inequality for the isotropic Cauchy distribution on the plane

T. Byczkowski
J. Małecki
T. Żak

Abstract: We prove an analogue of Ehrhard's inequality for the two-dimensional isotropic Cauchy measure. In contrast to the Gaussian case, the inequality is not valid for non-convex sets. We provide the proof for rectangles which are symmetric with respect to one coordinate axis.

2010 AMS Mathematics Subject Classification: Primary 60E05; Secondary 60E07.

Keywords and phrases: Cauchy distribution, Ehrhard-type inequality.

References


C. Borell, The Ehrhard inequality, C. R. Math. Acad. Sci. Paris 337 (2003), 663-667.

T. Byczkowski and T. Żak, Borell and Landau-Shepp inequalities for Cauchy-type measures, Probab. Math. Statist. 41 (2021), 129-152.

A. Ehrhard, Symétrisation dans l'espace de Gauss, Math. Scand. 53 (1983), 281-301.

R. J. Gardner, The Brunn-Minkowski Inequality, Bull. Amer. Math. Soc. 39 (2002), 355-405.

R. Latała and K. Oleszkiewicz, Gaussian measures of dilations of convex symmetric sets, Ann. Probab. 27 (1999), 1922-1938.

A. Prekopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. (Szeged) 32 (1971), 301-316.

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