About the optimal density associated to the chiral index of a sample from a bivariate distribution

Abstract

The complex quadratic form z ′ P z , where z is a fixed vector in C n and z ′ is its transpose, and P is any permutation matrix, is shown to be a convex combination of the quadratic forms z ′ P σ z , where P σ denotes the symmetric permutation matrices. We deduce that the optimal probability density associated to the chiral index of a sample from a bivariate distribution is symmetric. This result is used to locate the upper bound of the chiral index of any bivariate distribution in the interval [ 1 − 1 / π , 1 − 1 / 2 π ] . To cite this article: D. Coppersmith, M. Petitjean, C. R. Acad. Sci. Paris, Ser. I 340 (2005).