The finite-dimensional distributions of infinite-dimensional limit of finite-dimensional vectors

Question

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.

Answer 1

The question is a bit old but better late then never. Please have a look at the more recent MO question: Weak convergence for discrete-time processes using characteristic functions

Your question can be framed as a problem of weak convergence of probability measures on $\mathbb{R}^{\mathbb{N}}$ with the product topology. On this space, weak convergence is equivalent to that of the finite-dimensional marginals. So there is no "additional structure" you need to worry about.

As your notations are a bit awkward let me change them a bit. Let $\Sigma=(\Sigma_{st})_{s,t\in\mathbb{N}}$ denote the infinite matrix corresponding to the covariance of the input process $\varepsilon$ (with the Bourbaki-delinquent convention $\mathbb{N}=\{1,2,\ldots\}$), namely, $\Sigma_{st}=\mathbb{E}\ \varepsilon_s \varepsilon_t$.

Given your collection of matrices $C_k$ or rather $M^{(k)}=C_k^{-1}$ let me define the random variables $\theta^{(k)}$ in $\mathbb{R}^{\mathbb{N}}$ by $$ \theta_{s}^{(k)}=\sum_{a=1}^{k} M_{s,a}^{(k)}\varepsilon_a $$ for $1\le s\le k$ and by $\theta_{s}^{(k)}=0$ for $s>k$. Then you just need to make sure each entry of the infinite covariance matrix of the $\theta^{(k)}$ converges when $k\rightarrow\infty$. In other words the $\theta^{(k)}$ converge in distribution iff $$ \forall s,t\in\mathbb{N},\ \lim_{k\rightarrow\infty} \sum_{a,b=1}^{k}M_{s,a}^{(k)}M_{t,b}^{(k)}\Sigma_{a,b}\ {\rm exists}. $$ For instance you can take the $M^{(k)}$'s to be nested (not their inverses $C_k$) subject to the convergence of the series $$ \sum_{a,b=1}^{\infty}M_{s,a}M_{t,b}\Sigma_{a,b}\ . $$