A Gauss-Markov Theorem for Infinite Dimensional Regression Models with Possibly Singular Covariance - Morley, T. D. (SIAM journal on applied mathematics, 1979)
Consider the linear model z = Bx + v where x is a vector, B a linear operator, and v is a random variable with mean 0 and covariance$V^2 = \operatorname{cov} (v, v)$. A linear functional (g, ·) is called a best linear unbiased estimator for c if (i) B * g = c, and (ii) |V2g|2is as small as possible, subject to (i). The classical Gauss-Markov theorem gives a formula for the best linear unbiased estimator in finite dimensions, under the condition that V2is invertible. We extend the Gauss-Markov theorem to Hilbert space without the condition that the covariance is invertible.