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

Byczkowski Tomasz, Małecki Jacek, Żak Tomasz

Probability and Mathematical Statistics

2022

Vol. 42, Fasc. 1

163--175

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.

Borell and Landau–Shepp Inequalities for Cauchy-Type Measures

Byczkowski Tomasz, Żak Tomasz

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 1

129--152

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.