On Gaussian correlation inequalities for “periodic” sets

Abstract

Let A and B be convex sets in R 2 containing the origin which are invariant under rotation around the origin by a 2 π / k angle, k = 2 , 3 , 4 , 5 , … . In this paper we establish the correlation inequality P ( A ∩ B ) ⩾ P ( A ) P ( B ) under the N 2 ( 0 , I 2 ) distribution of X , for sets A and B as described above. This provides a generalization of Pitt's [1977. Ann. Probab. 5, 470–474] result, which established this correlation inequality for the case k = 2 , i.e. for convex symmetric sets.