A state-space calculus for rational probability density functions and applications to non-Gaussian filtering (Q2753227)

The use of linear system theory is proposed to analyse rational probability densities. For a rational probability density function \(\rho\) (not necessarily normalized) a rational function \(\Phi\) is defined on the imaginary axis by \(\Phi(ix)=\rho(x),\;x\in\mathbb{R}\) and it is extended to the complex plane. The additive decomposition of \(\Phi\) as \(\Phi(s)=Z(s)+Z^{*}(s),\;s\in\mathbb{C}\) is considered, where \(Z\) is a stable rational function and \(Z^{*}(s)=\overline{Z(-\overline{s}})\). The function \(Z\) is called the spectral summand of \(\Phi\) or the density summand of \(\rho\). The main tool in the state-space calculus developed in the paper is the interpretation of the density summand as the Fourier transform of the impulse response of a stable linear SISO system. The existence of moments of a random variable with unnormalized rational probability density function \(\rho\) is analysed by means of the Markov parameters of its density summand, and explicit formulae are derived for the existing moments in terms of these Markov parameters. State-space formulations are given for operations on rational probability densities, such as translation, scaling, product and convolution. This approach is used to treat the filtering problem in the case of a first-order linear stochastic model with independent noise variables and independent initial state with rational probability densities.