Let suppose to have a realization $\mathbf{X}=(\mathbf{X}_1,\dots, \mathbf{X}_n)$, where $\mathbf{X}_i \in \mathcal{R}^d$, from a $d-$variate Gaussian process.
Let also suppose that $E(\mathbf{X}_i)= \mathbf{0}_d$ and $Cov(\mathbf{X}_i)= \boldsymbol{\Sigma}$.
If I indicate with $C_{ij}$ the portion of the covariance matrix of $\mathbf{X}$ that rules the dependence between $\mathbf{X}_i$ and $\mathbf{X}_j$, and i assume that this must depend on the distance $|i-j|$, which are the necessary conditions needed to have the following? $$ f(\mathbf{X}_i|\mathbf{X}_{i-1},\mathbf{X}_{1-2},\dots,\mathbf{X}_1) = f(\mathbf{X}_i|\mathbf{X}_{i-1}) $$ where $f()$ is the normal density, i.e. the process is Markovian.
A reference with a proof is highly appreciated.
Thanks