Estimation and decision for linear systems with elliptical random processes

A random variable is said to have elliptical distribution if its density is a function of a quadratic form. This class includes the Gaussian and many other useful densities. It is shown in this paper that (a) this class of densities can be expressed as integrals of a set of Gaussian densities invariant under linear transformation; (b) the conditional expectation is linear with exactly the same form as the Gaussian case. Many estimation results of Gaussian case can be readily extended. Problems of computing optimal estimation, filtering, stochastic control and team decisions in various linear systems become tractable.