It is well known that the Wold's decomposition allows that every covariance-stationary time series $ Y_{{t}}$ can be written as the sum of two time series one deterministic $\eta _{t}$ and one stochastic$\sum _{j=0}^{\infty }b_{j}\varepsilon _{t-j}$.
$ Y_{t}=\eta _{t}+\sum _{j=0}^{\infty }b_{j}\varepsilon _{t-j}$
(see wikipedia for details)
If $\varepsilon _{t}$ is an uncorrelated sequence in terms of a white noise process it means that we are explaining $Y_{t}$ as a linear combination of an infinite number of white noise vectors, $\varepsilon _{t-i}$ for $i$ that goes to infinity, each one with variance finite and equal to the identity matrix.
My question is, how could this theorem be consistent with the idea that canonical Gaussian distribution in infinite dimensional Hilbert space does not exist?
Am I allowd to write the joint distribution of the infinite number of white noise vectors?