#StackBounty: #markov-process #covariance-matrix #gaussian-process Condition on the covariance matrix of a gaussian process needed to h…

Bounty: 50

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?
{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.


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