On Gaussian correlation inequalities for “periodic” sets

Let A and B be convex sets in R2 containing the origin which are invariant under rotation around the origin by a 2π/k angle, k=2,3,4,5,… . In this paper we establish the correlation inequality P(A∩B)⩾P(A)P(B) under the N2(0,I2) 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.