Spectral distribution function of a stationary time series ($F(\lambda )=\mathcal{F}([-\pi ,\lambda ])$)

The cumulative function of the finite positive spectral measure of a stationary time series is nondecreasing and right-continuous, with total mass equal to the variance. For an absolutely continuous measure, $F(\lambda )={\int }_{-\pi }^{\lambda }f(u)du$, where $f$ is the time-series spectral density. Under angular-frequency normalization, ${\gamma }_k={\int }_{-\pi }^{\pi }{e}^{ik\lambda }dF(\lambda )$ and, when the covariance series is absolutely summable, $f(\lambda )=(2\pi {)}^{-1}{\sum }_{k\in \mathbb{Z}}{\gamma }_k{e}^{-ik\lambda }$. Atoms describe persistent periodic components and cannot be represented by an ordinary density.