  Jointly optimal transmission power control and remote estimation over an
infinite horizon is studied. A sensor observes a dynamic process and sends its
observations to a remote estimator over a wireless fading channel characterized
by a time-homogeneous Markov chain. The successful transmission probability
depends on both the channel gains and the transmission power used by the
sensor. The transmission power control rule and the remote estimator should be
jointly designed, aiming to minimize an infinite-horizon cost consisting of the
power usage and the remote estimation error. A first question one may ask is:
Does this joint optimization problem have a solution? We formulate the joint
optimization problem as an average cost belief-state Markov decision process
and answer the question by proving that there exists an optimal deterministic
and stationary policy. We then show that when the monitored dynamic process is
scalar, the optimal remote estimates depend only on the most recently received
sensor observation, and the optimal transmission power is symmetric and
monotonically increasing with respect to the innovation error.
