On the Properties of Marginal Densities and Conditional Moments of Elliptically Contoured Measures

Abstract

We derive formulae relating marginal densities of an n-dimensional elliptically contoured vector y=(y₁,...,y_n) to the distribution of y^T ${\sum ^{ - 1}}y$, where $\sum { = EY{Y^T}}$ and to the conditional moments of the form:

$$ E\left( {y_1^{{j_1}} \ldots y_m^{{j_m}}\,{y_r}, \ldots, {y_n}} \right) $$

, where r > m and j₁,...,j_m are nonnegative integers.

It turns out that these densities and moments are defined by certain functions of one variable. We study such properties of these functions as different iability and monotonicity.

Using formulae defining conditional moments we characterize certain subclass of elliptically contoured measures. The elements of this subclass have especially simple form of some conditional moments and conditional correlation coefficient between the squares of say and y₁ and y₂.

This approach enables to look at Gaussian distribution as a sort of boundary distribution separating two, at first look different, subclasses of elliptically contoured measures.