I have read the following derivation in a book about correlation theory (Correlation theory of stationary and related random functions) and I need help understanding how the correlation function is derived.
The paper states that a random process can be established through convolving a function (typically smoothing kernel) $k(s)$ with another random process $x(s)$. Lets assume that $x(s)$ here is a Gaussian white noise process
$$y(s)=\int_{-\infty}^{\infty} \! k(u-s) x(u).du$$ Based on this the covariance is written as a function of $d=s-s'$ as follows $$Cov\left\{y(s),y(s')\right\}=E\left\{y(s)y(s') \right\}=$$ $$E\left\{\int_{-\infty}^{\infty} \! k(u-s) x(u)\,du\int_{-\infty}^{\infty} \! k(u'-s') x(u')\,du') \right\}= \int_{-\infty}^{\infty} \! k(u-d) k(u).du$$
The only conditions needed is that $x(s)$ is a continuous Gaussisan white noise process and that $k(s)$ is absolutely integrable.
I have been trying to understand how this equation is derived. Some points I was able to understand are
1) Integration here is possible since the continuous Gaussisan white noise is defined through a Dirac function $\delta(s)$ which is integrable
2) The stationarity assumption allows the covariance to be written in term of $d=s-s'$
I would really appreciate a detailed derivation to help me understand this approach for establishing random fields.