= Covariance criterion for a stationary Gaussian Markov process
{title2=$C(s+t)C(0)=C(s)C(t)$}
A centered continuous stationary scalar <Gaussian process> of positive variance can be <Markov> only if its covariance satisfies the displayed equation for $s,t\geq0$. Gaussian conditioning makes the future conditional mean $C(t)X_s/C(0)$; the <Markov property> then forces the equation by conditioning a future observation on the present and past. Conversely this equation makes the Gaussian residual future independent of the past, proving the <Markov property>. In the nondegenerate decaying case the covariance is a single exponential. A sum $c_1e^{-a_1|t|}+c_2e^{-a_2|t|}$ with positive coefficients and distinct positive decay rates fails the equation and hence is not scalar Markov.
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