= Mean-square derivative of a Gaussian process
{title2=$Y(t)=L^2\!\operatorname{-lim}_{h\to0}(X(t+h)-X(t))/h$}
A <Gaussian process> has a mean-square derivative at $t$ when its difference quotients converge in <L2 space>. The <mean-square convergence> need not give differentiable sample paths. For a centered <stationary process> with twice continuously differentiable <covariance function> $K(t-s)$, the difference quotients are Cauchy in <L2 space>, since their <covariance> is a rectangular average of $-K''$. Their limits form a <Gaussian process> with <covariance function> $\mathbb E[Y(s)Y(t)]=-K''(s-t)$.
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