I'm reading through a proof of the existence of a Brownian motion and at some point they state that for $0\leq t_{0}<t_{1}...<t_{n}$ there exist multivariate normal distributions with covariance matrices \begin{pmatrix} t_{0} & 0 & \cdots & \cdots & 0 \\ 0 & t_{1}-t_{0} & 0 & \cdots & 0 \\ 0 & 0 & t_{2}-t_{1} & \ddots & 0 \\ \vdots & \vdots & \ddots & \ddots & 0 \\ 0 & 0 & 0 & \cdots & t_{n}-t_{n-1} \end{pmatrix}
and
\begin{pmatrix} t_{0} & t_{0} & \cdots & \cdots & t_{0}\\ t_{0} & t_{1} & t_{1} & \cdots & t_{1}\\ t_{0} & t_{1} & t_{2} & \cdots & t_{2}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ t_{0} & t_{1} & t_{2} & \cdots & t_{n} \end{pmatrix}
I think the proof of this involves Kolmogorov's consistency theorem but I cannot quite see/proof why there exist multivariate normal distributions with such covariances. Could anyone help me see this? Your help would be much appreciated.
Kolmogorov's consistency theorem is not involved (and I am curious to know what gave you this idea). Rather one must either check that these matrices are symmetric definite nonnegative or realize some gaussian vectors with these covariance matrices from independent gaussian random vectors. Since both approaches are trivial in the first case, I am not sure about what is blocking you here.