If the target distribution is a Gaussian distribution, MALA (metropolis-adjusted langevin algorithm) becomes:

$$X_{t+1} = A X_t + \sqrt{2\tau}\xi$$ where $ A= I - \tau\Sigma^{-1}$.

where $\tau$ is a step size, $\Sigma$ is the covariance matrix of the target Gaussian distribution, and $\xi$ is an i.i.d. standard normal distribution.

I am wondering the following:

If we arbitrarily generate a matrix $A$ with the only constraint being convergent (i.e. can be non-positive definite), and run the dynamics shown in the first equation what would be the stationary distribution of $X_t$ would become at large $t$? More precisely, what would be the covariance matrix of the stationary distribution? Obviously, we cannot simply rearrange $ A= I - \tau\Sigma^{-1}$ and compute $ \Sigma = \tau (I-A)^{-1}$, since $ \Sigma $ wouldn't be positive definite in that case.



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