In this paper, we consider sampling an Ornstein-Uhlenbeck (OU) process through a channel for remote estimation. The goal is to minimize the mean square error (MSE) at the estimator under a sampling frequency constraint when the channel delay statistics is unknown. Sampling for MSE minimization is reformulated into an optimal stopping problem. By revisiting the threshold structure of the optimal stopping policy when the delay statistics is known, we propose an online sampling algorithm to learn the optimum threshold using stochastic approximation algorithm and the virtual queue method. We prove that with probability 1, the MSE of the proposed online algorithm converges to the minimum MSE that is achieved when the channel delay statistics is known. The cumulative MSE gap of our proposed algorithm compared with the minimum MSE up to the $(k+1)$-th sample grows with rate at most $\mathcal{O}(\ln k)$. Our proposed online algorithm can satisfy the sampling frequency constraint theoretically. Finally, simulation results are provided to demonstrate the performance of the proposed algorithm.

翻译：本文研究通过信道对奥恩斯坦-乌伦贝克过程进行采样以实现远程估计的问题。在信道延迟统计未知且受采样频率约束的条件下，目标是最小化估计器处的均方误差。我们将均方误差最小化下的采样问题重新表述为最优停时问题。通过回顾延迟统计已知时最优停时策略的阈值结构，提出一种在线采样算法，该算法利用随机逼近算法和虚拟队列方法学习最优阈值。我们证明：所提在线算法的均方误差以概率1收敛于信道延迟统计已知时的最小均方误差；与最小均方误差相比，到第$(k+1)$个样本为止的累积均方误差差距以不超过$\mathcal{O}(\ln k)$的速率增长。所提在线算法理论上可满足采样频率约束。最后，通过仿真结果验证所提算法的性能。