WeaklyStationaryCovariance

The covariance function of a weakly stationary stochastic process is characterized by the following properties:

1. Dependence on Time Interval: The covariance function $R(t, s)$ depends only on the time interval $\tau = t - s$, not on the specific times $t$ and $s$. This is expressed as:

$$R(t, s) = R(t - s) = R(\tau)$$

2. Symmetry: The covariance function is symmetric with respect to the time interval, meaning $R(\tau) = R(-\tau)$.

3. Finite and Constant Variance: The variance of the process at any time point is finite and constant over time, indicated by $R(0)$ being finite.

4. Spectral Representation: The covariance function $R(\tau)$ can be represented as the Fourier transform of a non-negative, finite measure $F(\lambda)$, known as the spectral distribution function:

$$ R(\tau) = \int_{-\infty}^{\infty} e^{i\lambda\tau} , dF(\lambda) $$

This representation, which is a consequence of Bochner's theorem, ensures that the covariance function is positive definite and encapsulates the frequency content of the process.

In summary, the covariance function of a weakly stationary stochastic process is characterized by its dependence only on the time interval between two points, its symmetry, finite constant variance, and a spectral representation that adheres to Bochner's theorem. These properties are useful for facilitating the analysis of autocorrelation functions and power spectra among other things.