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Let $X_k = A_{k-1}X_{k-1} + \omega_{k-1}$, and $Y_k = H_{k} X_k +\eta_k$, where $\omega_k$ and $\eta_k$ are i.i.d. Gaussian random variables. We now can get unbiased estimate of the random variable $X_k$ through $\hat X_k = (H_k)^{\dagger}Y_k$ assuming $H_k$ has full column rank, where $(H_k)^\dagger$ represent a pseudo inverse.

Q: Is the matrix $\hat X_k\hat X^T_k$ ergodic?

It feels like a no. but the simulations tell me the time averages are still good estimates of the ensemble averages.

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    $\begingroup$ How can a constant be ergodic? This is a property of a stochastic process. $\endgroup$ – Taylor Nov 5 '17 at 1:11
  • $\begingroup$ Thank you for pointing it out. It should be $\hat X_k\hat X^T_k$ instead of $\mathbf{E}[\hat X_k\hat X^T_k]$. $\endgroup$ – ZHUANG Nov 5 '17 at 5:06

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