5.4 Spherical and Elliptical Distributions

The multinormal distribution belongs to the large family of elliptical distributions which has recently gained a lot of attention in financial mathematics. Elliptical distributions are often used, particularly in risk management.

DEFINITION 5.1   A random vector is said to have a spherical distribution if its characteristic function satisfies: for some scalar function which is then called the characteristic generator of the spherical distribution . We will write .

This is only one of several possible ways to define spherical distributions. We can see spherical distributions as an extension of the standard multinormal distribution .

THEOREM 5.10   Spherical random variables have the following properties:
1. All marginal distributions of a spherical distributed random vector are spherical.
2. All the marginal characteristic functions have the same generator.

3. Let , then has the same distribution as where is a random vector distributed uniformly on the unit sphere surface in and is a random variable independent of . If , then

The random radius is related to the generator by a relation described in Fang et al. (1990, p.29). The moments of , provided that they exist, can be expressed in terms of one-dimensional integrals (Fang et al.; 1990).

A spherically distributed random vector does not, in general, necessarily possess a density. However, if it does, the marginal densities of dimension smaller than are continuous and the marginal densities of dimension smaller than are differentiable (except possibly at the origin in both cases). Univariate marginal densities for greater than are nondecreasing on and nonincreasing on .

DEFINITION 5.2   A random vector is said to have an elliptical distribution with parameters and if has the same distribution as , where and is a matrix such that with . We shall write .

REMARK 5.1   The elliptical distribution can be seen as an extension of .

EXAMPLE 5.6   The multivariate t-distribution. Let and be independent. The random vector has a multivariate -distribution with degrees of freedom. Moreover the -distribution belongs to the family of p-dimensioned spherical distributions.

EXAMPLE 5.7   The multinormal distribution. Let . Then and . Figure 4.3 shows a density surface of the multivariate normal distribution: with and Notice that the density is constant on ellipses. This is the reason for calling this family of distributions elliptical''.

THEOREM 5.11   Elliptical random vectors have the following properties:
1. Any linear combination of elliptically distributed variables are elliptical.

2. Marginal distributions of elliptically distributed variables are elliptical.

3. A scalar function can determine an elliptical distribution for every and with iff is a -dimensional characteristic function.

4. Assume that is nondegenerate. If and , then there exists a constant such that

In other words are not unique, unless we impose the condition that .

5. The characteristic function of is of the form

for a scalar function .

6. with iff has the same distribution as:

where is independent of which is a random vector distributed uniformly on the unit sphere surface in and is a matrix such that .

7. Assume that and . Then

8. Assume that with . Then

has the same distribution as in equation (5.18).