This paper addresses least-squares estimation of parameters in digital input/output models of linear time-invariant distributed systems in the presence of white process and sensor noise. The systems of interest have state-space realizations in Hilbert spaces. Both finite-dimensional and infinite-dimensional input/output models are considered. The paper derives a number of new results for recursive least-squares estimation and filtering. The main results characterize the asymptotic values to which parameter estimates converge with increasing amounts of data. The most important result is an equivalence between least-squares parameter estimation on an infinite interval (i.e., with infinitely long data sequences) and linear-quadratic optimal control on a finite interval. Numerical results are presented for a sampled-data version of a wave equation.