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## The finite-dimensional distributions of infinte-dimensional limit of finite-dimensional vectors

Suppose we have the process $\{\varepsilon_t,t\in \mathbb{N}\}$. Suppose that this the finite-dimensional distributions of this process are Gaussian, i.e. for any $t_1,...,t_n$, vector $(\varepsilon_{t_1},...,\varepsilon_{t_n})$ is multivariate normal with zero mean and covariance matrix $\Sigma_{t_1,...,t_n}$.

Suppose we have a sequence of matrices $\{C_k,k\in \mathbb{N}\}$. For each $k$ matrix $C_k$ is a positive-definite $k\times k$ matrix. Define $\theta_k=C_k^{-1}{\varepsilon}^k$, where $\varepsilon^k=(\varepsilon_1,...,\varepsilon_k)'$.

Denote the $l$-th element of $\theta_k$ by $\theta^k_l$. I am interested in the limits (in distribution)

$$\theta^l=\lim_{k\to\infty}\theta^k_l$$

Do we need any additional requirements for the $\varepsilon$ and $C_k$ for the existence of the limit?

If the limits exists what can we say about the finite-dimensional distributions of process $\{\theta^l,l\in \mathbb{N}\}$?

We can suppose that matrices $C_k$ are nested, i.e. the submatrix of matrix $C_{k+1}$ containing first $k$ rows and $k$ columns is matrix $C_k$.

Somehow I feel the problem might be ill-posed and probably I need additional structure besides finite-dimensional distributions. Any pointers of what I am doing wrong would be very welcome.

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