The Wiener–Khinchin theorem says the autocorrelation function of a wide sense stationary process can be written as a Stieltjes integral, where the integrator function is called the power spectral distribution function. When the power spectral distribution function is absolutely continuous, its derivative is called the power spectral density, and the power spectral density and the autocorrelation function are a Fourier transform pair.
My question is when and how the power spectral density of the stationary stochastic process can be represented as in Wikipedia: for a stationary process $x(t), t \geq 0$,
the power spectral density can be defined as $$ S_{xx}(\omega) := \lim_{T \rightarrow \infty} \mathbf{E} \left[ | \frac{1}{\sqrt{T}} \int_0^T x(t) e^{-i\omega t}\, dt | ^ 2 \right]. $$
Thanks and regards!