A State-Space Calculus for Rational Probability Density Functions and Applications to Non-Gaussian Filtering

We propose what be believe to be a novel approach to perform calculations for rational density functions using state space representations of the densities. By standard results from realisation theory, a rational probability density function is considered to be the transfer function of a linear system with generally complex entries. The stable part of this system is positive-real, which we call the density summand. The existence of moments is investigated using the Markov parameters of the density summand. Moreover, explicit formulae are given for the existing moments in terms of these Markov parameters. One of the main contributions of the paper are explicit state space descriptions for products and convolutions of rational densities. As an application which is of interest in its own right, the filtering problem is investigated for a linear time-varying system whose noise inputs have rational probability density functions. In particular state space formulations are derived for the calculation of the prediction and update equations. The case of Cauchy noise is treated as an illustrative example.