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.

翻译：本文分析远程估计系统中信息老化的单调性，该系统利用高斯自回归AR(p)过程的历史观测值预测其未来值。我们考虑估计中两种常用的损失函数：(i) 用于极大似然估计的对数损失函数，(ii) 用于最小均方误差估计的二次损失函数。AR(p)过程的估计误差表示为具有闭式解的广义条件熵。通过运用名为$\epsilon$-马尔可夫链的新型信息论工具，我们可评估AR(p)过程偏离马尔可夫链的程度。当偏离度$\epsilon$较大时，AR(p)过程的估计误差可能远非信息年龄（AoI）的非递减函数；反之，当偏离度$\epsilon$较小时，推断误差接近AoI的非递减函数。每个观测值均为取自AR(p)过程的短序列。随着观测序列长度增加，参数$\epsilon$逐渐趋近于零，从而使估计误差成为AoI的非递减函数。这些结果揭示了信息老化单调性与过程偏离马尔可夫链程度之间的内在联系。