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Use the standard linear Gaussian state-space model convention: the initial state is independent of both mutually independent iid noise sequences. This independence from the initial state is needed for the usual forecast recursion. Let ${\mathcal{Y}}_t=\sigma (y_1,\dots ,y_t)$ and write $m_{t-1}=x_{t-1}^{t-1}$ and $C_{t-1}=P_{t-1}^{t-1}$. Taking conditional expectations in the state equation givesTo justify the covariance, the forecast error is $\Phi (x_{t-1}-m_{t-1})+w_t$. Its two terms have zero cross-covariance because $w_t$ is independent of the previous state and observations. Their covariances therefore add. In this Gaussian model the conditional error covariance depends on the parameters and time but not on the realized data, so it also equals the unconditional forecast-error covariance used in the notation of the question.

For completeness, derive the filtering step rather than leave the algorithm unspecified. Set $m_t^{-}=x_t^{t-1}$ and $C_t^{-}=P_t^{t-1}$. Conditionally on the previous data, the observation has mean $A{m}_t^{-}$ and covarianceDefine the innovation process and gain byThe state forecast error and innovation are jointly normal, with cross-covariance $C_t^{-}{A}^{\top }$. The residual $x_t-m_t^{-}-K_t{ϵ}_t$ is uncorrelated with ${ϵ}_t$ by the definition of $K_t$, hence independent of it by joint normality. Its mean is zero and its covariance is $C_t^{-}-C_t^{-}{A}^{\top }{\Sigma }_t^{-1}A{C}_t^{-}$. This proves the conditional multivariate normal distribution updateInitialize $x_0^0={\mu }_0,P_0^0={\Sigma }_0$. For $t=1,\dots ,n$, predict using the first boxed pair, store that forecast, form ${ϵ}_t,{\Sigma }_t,K_t$, and update using the second boxed pair. The updated state then starts the next iteration. This is the Kalman filter, and supplies the entire forecast sequence in one forward pass.

The requested unconditional innovation moments areIn fact ${ϵ}_t\mid {\mathcal{Y}}_{t-1}\sim N_q(0,{\Sigma }_t)$, with ${\Sigma }_t$ deterministic for fixed parameters. Thus the innovation is independent of the past observations, and hence the innovations are mutually independent. Equivalently, orthogonality to the past together with joint normality gives this independence.

The chain rule for densities now yields the Gaussian innovation likelihood, treating the specified initial mean and covariance as known:Both ${ϵ}_t$ and ${\Sigma }_t$ are functions of the trial parameters, so the filtering recursion must be run again for each candidate $\Theta$. The ordinary density formula assumes ${\Sigma }_t$ positive definite; for example positive definite $R$ ensures this. Singular observation laws instead require densities on their Gaussian support.

Evaluate the log-likelihood numerically by the forward recursion, and maximize it over admissible $\Phi ,A,Q,R$, enforcing the covariance constraints. A Cholesky decomposition ${\Sigma }_t=L_t{L}_t^{\top }$ gives $\log |{\Sigma }_t|=2{\sum }_j\log (L_t{)}_{jj}$ and the quadratic term $\Vert L_t^{-1}{ϵ}_t{\Vert }^2$, avoiding an explicit matrix inverse. Covariance parameterizations or constrained optimization keep $Q,R$ positive semidefinite and the innovation covariance nonsingular. An attained maximizer is a maximum-likelihood estimator; in general this is a numerical optimization problem and neither uniqueness nor global convergence of a local optimizer is automatic.