  In this paper, we analyze the monotonicity of information aging in a remote
estimation system, where historical observations of a Gaussian autoregressive
AR(p) process are used to predict its future values. We consider two widely
used loss functions in estimation: (i) logarithmic loss function for maximum
likelihood estimation and (ii) quadratic loss function for MMSE estimation. The
estimation error of the AR(p) process is written as a generalized conditional
entropy which has closed-form expressions. By using a new information-theoretic
tool called $\epsilon$-Markov chain, we can evaluate the divergence of the
AR(p) process from being a Markov chain. When the divergence $\epsilon$ is
large, the estimation error of the AR(p) process can be far from a
non-decreasing function of the Age of Information (AoI). Conversely, for small
divergence $\epsilon$, the inference error is close to a non-decreasing AoI
function. Each observation is a short sequence taken from the AR(p) process. As
the observation sequence length increases, the parameter $\epsilon$
progressively reduces to zero, and hence the estimation error becomes a
non-decreasing AoI function. These results underscore a connection between the
monotonicity of information aging and the divergence of from being a Markov
chain.
