# #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?
$$f(mathbf{X}_i|mathbf{X}$$
{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

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