The spectrum of \(X\) is the set of \[ \{ x: F(x+\varepsilon)-F(x-\varepsilon)>0; \quad\forall \varepsilon>0 \}. \] If \(F(\cdot)\) is absolutely continuous with respect to the [[lebesgue-measure|Lebesgue measure]], i.e. \(F(\cdot)=\int_{-\infty}^\cdot (\frac{dF(s)}{ds})ds\) then its Radon-Nikodym Derivative is called the spectral density.
Suppose that \(\{ X_{t}, t \in \mathbb{R} \}\) is weakly stationary with mean \(0\) and variance \(1\) and with spectral distribution \(F(\cdot)\). Then we have the spectral representation \[ X_{t}=\int _{\mathbb{R}}e^{itu}dZ(u), \] where the RHS is the limit in mean square of a sequence of approximating Riemann-Stieltjes sums and the spectral process \(\{ Z(u),u \in \mathbb{R} \}\) is a complex-valued process of orthogonal increments, i.e. \[ \mathbb{E}[(Z(u_{4})-Z(u_{3})(\overline{Z(u_{2})-Z(u_{1})})]=0;\quad u_{1}<u_{2}\leq u_{3}<u_{4}, \] with \[ \mathbb{E}[(Z(u_{2})-Z(u_{1}))(\overline{Z(u_{2})-Z(u_{1})})]=F(u_{2}-F(u_{1})),\quad u_{1}<u_{2}. \] Moreover, we have for every \(u_{1}<u_{2}\) the inversion formulae \[ \begin{align}F(u_{2})-F(u_{1})&=\lim_{T \to \infty} \frac{1}{2 \pi}\int_{-T}^T \frac{e^{-iu_{2}t}-e^{-iu_{1}t}}{-it}\cdot \rho(t)\;dt,\\ Z(u_{2})-Z(u_{1})&=\text{l.i.m.}_{T \to \infty} \frac{1}{2\pi}\int_{-T}^T \frac{e^{-iu_{2}t}-e^{-iu_{1}t}}{-it}\cdot X_{t}\;dt,\end{align} \] where \(\text{l.i.m}\) is the limit in mean square of the sequence of random variables.
If \(\{ X_{n},n \in \mathbb{Z} \}\) is a complex-valued, discrete-time weakly stationary process with zero mean and unit variance and with the autocorrelation function \(\rho(n),~n \in \mathbb{Z}\). There exists the spectral distribution function \(F:(-\pi,\pi] \to [0,1]\) such that \[ \rho(n)=\int_{-\pi}^\pi e^{inu}dF(u);\quad n \in \mathbb{Z}. \] If it has the spectral density function \(f(\cdot)\), then \[ f(u)= \frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{-i\nu}\rho(n);\quad u \in [-\pi,\pi). \]