Optimal Sampling and Scheduling for Remote Fusion Estimation of Correlated Wiener Processes

In distributed sensor networks, sensors often observe a dynamic process within overlapping regions. Due to random delays, these correlated observations arrive at the fusion center asynchronously, raising a central question: How can one fuse asynchronous yet correlated information for accurate remote fusion estimation? This paper addresses this challenge by studying the joint design of sampling, scheduling, and estimation policies for monitoring a correlated Wiener process. Though this problem is coupled, we establish a separation principle and identify the joint optimal policy: the optimal fusion estimator is a weighted-sum fusion estimator conditioned on Age of Information (AoI), the optimal scheduler is a Maximum Age First (MAF) scheduler that prioritizes the most stale source, and the optimal sampling can be designed given the optimal estimator and the MAF scheduler. To design the optimal sampling, we show that, under the infinite-horizon average-cost criterion, optimizing AoI is equivalent to optimizing MSE under pull-based communications, despite the presence of strong inter-sensor correlations. This structural equivalence allows us to identify the MSE-optimal sampler as one that is AoI-optimal. This result underscores an insight: information freshness can serve as a design surrogate for optimal estimation in correlated sensing environments.