# A note on conditional covariance matrices for elliptical distributions

## TL;DR

This paper derives an analytical formula for conditional covariance matrices of elliptically distributed vectors, simplifying calculations and exploring properties of quantile-based sets under multivariate normality.

## Contribution

It introduces a univariate invariant that simplifies the calculation of conditional covariance matrices for elliptical distributions and characterizes quantile-based sets with equal conditional covariances.

## Key findings

- Derived an explicit formula for conditional covariance matrices.
- Introduced a univariate invariant for simplification.
- Characterized quantile-based sets with equal conditional covariances.

## Abstract

In this short note we provide an analytical formula for the conditional covariance matrices of the elliptically distributed random vectors, when the conditioning is based on the values of any linear combination of the marginal random variables. We show that one could introduce the univariate invariant depending solely on the conditioning set, which greatly simplifies the calculations. As an application, we show that one could define uniquely defined quantile-based sets on which conditional covariance matrices must be equal to each other if only the vector is multivariate normal. The similar results are obtained for conditional correlation matrices of the general elliptic case.