We consider an estimation problem in which the signal is modelled by a continuous Gaussian random field and is observed through smooth and bounded nonlinear sensors. A nonhomogeneous Markov process is defined in order to sample the conditional distribution of the signal given the observations. At any finite time the process takes values in a finite-dimensional space, although the dimension goes to infinity in time. We prove that the empirical averages of any bounded functional continuous w.p.1 converge in the mean square to the conditional expectation of the functional.
"A Nonhomogeneous Markov Process for the Estimation of Gaussian Random Fields with Nonlinear Observations." Ann. Probab. 19 (4) 1664 - 1678, October, 1991. https://doi.org/10.1214/aop/1176990228