  The fundamental limits of remote estimation of Markov processes under
communication constraints are presented. The remote estimation system consists
of a sensor and an estimator. The sensor observes a discrete-time Markov
process, which is a symmetric countable state Markov source or a Gauss-Markov
process. At each time, the sensor either transmits the current state of the
Markov process or does not transmit at all. Communication is noiseless but
costly. The estimator estimates the Markov process based on the transmitted
observations. In such a system, there is a trade-off between communication cost
and estimation accuracy. Two fundamental limits of this trade-off are
characterized for infinite horizon discounted cost and average cost setups.
First, when each transmission is costly, we characterize the minimum achievable
cost of communication plus estimation error. Second, when there is a constraint
on the average number of transmissions, we characterize the minimum achievable
estimation error. Transmission and estimation strategies that achieve these
fundamental limits are also identified.
