Spectrum antitrasformation and relation with covariance

There exists an antitrasformation: $$\gamma_y(\tau)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}{\Gamma_y(\omega)e^{j\omega\tau}d\omega}$$

Notice that the variance: $$\gamma_y(0)=\mathbb{E}[(y(t)-m_y)^2]=\frac{1}{2\pi}\int_{-\pi}^{+\pi}{\Gamma_y(\omega)d\omega}$$ so the variance is the area below the spectrum.

There is biunivocal relationship between spectrum and covariance function. This means that the weak (wide sense) characterization of a process is given by the mean function and by the spectrum or the covariance function.

The spectrum and the covariance function contain the same information under a different prospective.