Let $\varepsilon_{s}^{FR}$ follow a Gaussian Markov process so that ${\varepsilon_{s}^{FR} = \rho \varepsilon_{s-1}^{FR} + \xi_{s}^{FR}, \: \xi_{s}^{FR} \sim \mathrm{i.i.d.} \: N(0,\sigma^{2}_{\xi^{FR}})}$ where the initial value, $\varepsilon_{t}^{FR}$ is i.i.d. $N(0,\sigma^{2}_{\varepsilon^{FR}})$ and is independent of $\xi_{s}^{FR}$, $s = t+1, \dotsc, S$. It follows that the vector $\varepsilon_{t}^{FR}, \dotsc, \varepsilon_{\tau}^{FR}$ is multivariate normal distributed with mean zero, variance $\rho^{2(s-t)}\sigma_{\varepsilon^{FR}}^{2} + \sum_{i=0}^{s-t-1} \rho^{2i}\sigma^{2}_{\xi^{FR}}$, and covariance $\rho^{s-t} \sigma_{\varepsilon_{t}^{FR}}^{2}$. (This last claim is from a published article.) Here $\tau \in \{t, \dotsc,T\}$. I denote this mean by $\Omega^{FR}$ and the covariance matrix with $\Sigma^{FR}$.

Let $\varepsilon_{s}^{PR}$ be another process and it follows a Gaussian Markov process just like $\varepsilon_{s}^{FR}$ with error $\xi_{s}^{PR}$ with mean $\Omega^{PR}$ and covariance matrix $\Sigma^{PR}$, and hence the vector $\varepsilon_{t}^{PR}, \dotsc, \varepsilon_{\tau}^{PR}$ is multivariate normal distributed too. I assume that the $\rho$ paramater is same in $\varepsilon_{s}^{FR}$ and $\varepsilon_{s}^{PR}$ (I could assume that each process has a different $\rho$ but this is not central to my question).

Furthermore, let that $\varepsilon_{t}^{FR}$ is defined as (or determined by) $\upsilon_{t} - \omega_{t}$, and $\varepsilon_{t}^{PR}$ is defined as $\mu_{t} - \omega_{t}$. Hence, $\varepsilon_{t}^{FR}$ and $\varepsilon_{t}^{PR}$ are contemporaneously correlated through $\omega_{t}$.

Let $\upsilon_{t}$, $\omega_{t}$ and $\mu_{t}$ all follow a Markovian process such that $\upsilon_{s} = \rho \upsilon_{s-1} + \phi_{s}$, $\omega_{s} = \rho \omega_{s-1} + \psi_{s}$, and $\mu_{s} = \rho \mu_{s-1} + \gamma_{s}$, where I assume that $E_{s-1}(\phi_{s}) = 0$, $E_{s-1}(\psi_{s}) = 0$, and $E_{s-1}(\gamma_{s}) = 0$ for $s = t+1, \dotsc, S$.

This shows that I have two multivariate normal distributions (i.e. the AR(1) processes $\varepsilon_{s}^{FR}$ and $\varepsilon_{s}^{FR}$) that are (contemporenously) correlated. How is then the vector $\varepsilon_{s}^{FR}$, $\varepsilon_{s}^{PR}$ distributed? Is it multivariate normally distributed?

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    $\begingroup$ I am very much seeking an answer to proceed with my research. Even some tips might be very helpful. $\endgroup$ – Snoopy Feb 29 '16 at 19:43

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