In this paper we consider the problem of infinite-horizon sensor scheduling for estimation in linear Gaussian systems. Due to possible channel capacity, energy budget or topological constraints, it is assumed that at each time step only a subset of the available sensors can be selected to send their observations to the fusion center, where the state of the system is estimated by means of a Kalman filter. Several important properties of the infinite-horizon schedules will be presented in this paper. In particular, we prove that the infinite-horizon average estimation error and the boundedness of a schedule are independent of the initial covariance matrix. We further provide a constructive proof that any feasible schedule with finite average estimation error can be arbitrarily approximated by a bounded periodic schedule. We later generalized our result to lossy networks. These theoretical results provide valuable insights and guidelines for the design of computationally efficient sensor scheduling policies.