Stationary stochastic process

Motivation

The main advantage of wide-sense stationarity is that it places the time-series in the context of Hilbert spaces. Let ⁠${\displaystyle H}$⁠ be the Hilbert space generated by ⁠${\displaystyle \{x(t)\}}$⁠ (that is, the closure of the set of all linear combinations of these random variables in the Hilbert space of all square-integrable random variables on the given probability space). By the positive definiteness of the autocovariance function, it follows from Bochner's theorem that there exists a positive measure ${\displaystyle \mu }$ on the real line such that ⁠${\displaystyle H}$⁠ is isomorphic to the Hilbert subspace of ⁠${\displaystyle L^{2}(\mu )}$⁠ generated by ⁠${\displaystyle \{e^{-2\pi i\xi \cdot t}\}}$⁠. This then gives the following Fourier-type decomposition for a continuous time stationary stochastic process: there exists a stochastic process ${\displaystyle \omega _{\xi }}$ with orthogonal increments such that, for all ⁠${\displaystyle t}$⁠: