Suppose X(t),$t\in R$ is a stationary Gaussian process,with covariance kernel R(s,t)=R(s-t) (abusing the notation a little) analytic. Can we say anything about the covariance kernel of the Gaussian process $X^1(t)$? (where $f^1(t)$ denotes $\frac{d}{dt}f(t)$.
Covariance Kernel of $X^1(t)$,where X(t) is a stationary Gaussian Process
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calculus
stochastic-processes
covariance
1 Answers
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For the $L^2$-differentiation of second order processes(Gaussian process is a second order process), there is following facts(c.f. M. Loeve, Probability Theory II, 4th. Ed., Spring Verlag, 1978, 37.2.c, pp.136--): Let $\{X(t)\}$ be a wide-stationary process with $\mathsf{E}[X(t)]=0$, $\mathsf{E}[X(t)X(s)]=B(t-s)$ and $$\lim_{h\to0}\mathsf{E}\biggl|\frac{X(t+h)-X(t)}{h}-X'(t)\biggr|^2=0.$$ Then $$\mathsf{E}{X'(t)X'(s)}=-B''(t-s).$$