Some convergence theorems for linear systems

Given a probability space ; -component complex-valued random variables on a positive integer, whose components have finite variance and mean zero; and two sequences of matrices where the are and the are positive integers, matrices whose elements are complex constants. We consider the stochastic process where begin{equation} X_{n+1} = phi_n X_n + V_n qquad n geq 1 end{equation} and the associated sampling procedure begin{equation} Y_n = M_n X_n qquad n geq 1. end{equation} We pose the following question. If in the "prediction problem" and the "filtering problem" we let and , respectively, be the "best estimates," under what circumstances do the differences and - "approach 0 as ". In Secion I we give definitions and some properties. In Section II, we give an answer in the special case where and (i.e., they are independent of ). In Section III we prove a theorem that gives a deeper insight into the condition under which the theorem proved in Section II holds.