Consider the $p\times p$ matrix $\pmb{A}$ and suppose $\theta_n, n\ge 0$ are $p$-dimensional vectors we want to estimate according to MMSE criterion.
We have the recursive relation $\theta_n = \pmb{A} \theta_{n-1}, n\ge 1$. Let $\hat{\theta}_0$ be the MMSE estimator of $\theta_0$. Then, since MMSE estimator commutes over affine mappings, the MMSE estimator of $\theta_n$ is given as $\hat{\theta}_n = \pmb{A}^n \hat{\theta}_0$.
Can we still find the MMSE estimate of $\theta_n$ via $\hat{\theta}_n = \pmb{A}^n \hat{\theta}_0$ if $\pmb{A}$ is not invertible?