Telescoping Recursive Representations and Estimation of Gauss-Markov Random Fields

We present telescoping recursive representations for both continuous and discrete indexed noncausal Gauss-Markov random fields. Our recursions start at the boundary (for example, a hypersurface in R, d ≥ 1) and telescope inwards. Under appropriate conditions, the recursions for the random field are differential/difference representations driven by white noise, for which we can use standard recursive estimation algorithms, such as the Kalman-Bucy filter and the Rauch-Tung-Striebel smoother. Index Terms Random Fields, Gauss-Markov Random Fields, Reciprocal Processes, Gauss-Markov Random Processes, Kalman Filter, Rauch-Tung-Striebel Smoother, Recursive Estimation, Recursive Processing, Telescoping Representation