A note on the convergence of the responses of stochastic systems (Q2735301)

This is a note on the convergence theorem for the discrete-time asymptotically stable dynamical system \(y(t)= \sum^\infty_{\tau=0} h(\tau)u (t-\tau)\), \(t\in\mathbb{Z}\), where \(u\) and \(y\) are the stochastic input and stochastic output (SISO), respectively, as given in [\textit{K. J. Aström}, Introduction to stochastic control theory. New York: Academic Press (1970; Zbl 0226.93027), Theorem 2.1, p. 94]. Via an example, it is pointed out that there exists a negligence and the original condition about the input for the convergence theorem, i.e., that the input \(\{u(t)\}\) is a stochastic process of second order with mean and covariance, should be replaced by the condition that \(\{u(t)\}\) is of strong second order, i.e., \(\forall t\in\mathbb{Z}\), \(\sup_{k\leq t}E|u(t) |^2 <\infty\). A proof for the revised convergence theorem is also given.