From Wikipedia,
there exists a stochastic process $\omega _{\xi }$ with orthogonal increments such that, for all $t$, the weakly stationary process $X_t = \int e^{- 2 \pi i \lambda \cdot t} \, d \omega_\lambda,$
It is deducted by applying Bochner's theorem on the autocovariance function which is positive-definite. However, I don't understand how the stationary process could also be regarded as a Fourier-transform.
Still from Wikipedia,
By the positive definiteness of the autocovariance function, it follows from Bochner's theorem that there exists a positive measure $ > \mu $ on the real line such that $H$ is isomorphic to the Hilbert subspace of $L^2(μ)$ generated by $e^{−2πiξ⋅t}$. This then gives the following Fourier-type decomposition for a continuous time stationary stochastic process.
The same result holds for a discrete-time stationary process, with the spectral measure now defined on the unit circle.
Could someone explain it with more details?