Convergence of eigenvalues in state-discretization of linear stochastic systems

The transition operator that describes the time evolution of the state prob ability distribution for continuous-state linear systems is given by an int egral operator. A state-discretization approach is proposed, which consists of a finite rank approximation of this integral operator. As a result of t he state-discretization procedure, a Markov chain is obtained, in which cas e the transition operator is represented by a transition matrix. Spectral p roperties of the integral operator for the continuous-state case are presen ted. The relationships between the integral operator and the finite rank ap proximation are explored. In particular, the limiting properties of the eig envalues of the transition matrices of the resulting Markov chains are stud ied in connection to the eigenvalues of the original continuous-state integ ral operator.