# consistency of EM estimation in Kalman filter

I do not understand the properties of EM algorithm of Kalman.

Assume one has a state-space model: $$z_{t} = B x_{t} + v_{k}\\ x_{t+1} = A x_{t} + w_{k},$$ where noise terms are i.i.d. such that $$v_{k} \sim N(0, Q)$$ and $$w_{k} \sim N(0, R)$$. Assume that the matrices $$A$$ and $$B$$ are known.

The question: does EM algorithm provide a consistent estimation of $$Q$$ and $$R$$, the observation and transition covariances?

My discovery:

and some simple simulations, the estimate of the covariance matrix is far from the true values.

From the discussion here: https://dsp.stackexchange.com/questions/24417/em-algorithm-and-kalman-filter

However, EM algorithm has a drawback. It only gives suboptimal solution. In other words, the parameters estimated by EM algorithm are only local minimum/maximum, rather than global minimum/maximum. Therefore, further turning may be needed.

Therefore, my conclusion is that in general Kalman EM does not provide a consistent estimator of the variance.