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In this paper, we analyze the monotonicity of infor-mation 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 ϵ -Markov chain, we can evaluate the divergence of the AR(p) process from being a Markov chain. When the divergence ϵ 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 ϵ, the estimation 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 ϵ 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.