I'm studying stochastic process and Markov Chain.
I was wondering if a Gaussian Process has the Markov Property (that is the conditional probability distribution (given the present states) of future states is independent of the past states). Personally, following my intuition, I would say that a Gaussian Process has the Markov Property only when the covariance of the Gaussian is a diagonal matrix.
My reasoning is the following: A Gaussian Process with a diagonal covariance matrix is a process of independently distributed random variables, and so by definition has the Markov Property.
Is my reasoning correct? Is the assumption of diagonal covariance necessary?