State space calculations for discrete probability densities

Abstract

State space methods have proved to be powerful theoretical and computational tools in a number of areas of applications, in particular filtering and control theory. In this paper we advocate the use of state space methods for the study of discrete probability densities on the set {0,1,2,…}. The fundamental approach is to consider the class D of all discrete probability densities that can be represented as the impulse response/convolution kernel of a finite dimensional discrete time state space system. We show that all standard operations such as the calculation of moments, convolution, scaling, translation, product, etc. can be carried out using system representations.