Stationarity

A random process is strict-sense stationary if the joint density function of the random variables obtained by sampling that process is invariant under arbitrary time shifts:

$$f_{X(t_1), \dots X(t_\ell)}(x_1, \dots, x_\ell) = f_{X(t_1 + \alpha), \dots X(t_\ell + \alpha)}(x_1, \dots, x_\ell)$$

A random process is strict-sense stationarity if:

• The mean $\mu_X(t)$ is invariant with time:

$$\mu_X(t) = \mu_X$$

• The autocorrelation $R_{XX}(t_1, t_2)$ and autocovariance $C_{XX}(t_1, t_2)$ only depend on the time difference $(t_1 - t_2)$.

$$R_{XX}(t_1, t_2) = R_{XX}(t_1 - t_2)$$ $$C_{XX}(t_1, t_2) = C_{XX}(t_1 - t_2)$$

Strict-sense stationarity implies wide-sense stationarity.

Symmetry properties:

$$R_{xx}(\tau) = R_{xx}(-\tau)$$ $$C_{xx}(\tau) = C_{xx}(-\tau)$$

$$R_{xy}(\tau) = R_{yx}(-\tau)$$ $$C_{xy}(\tau) = C_{yx}(-\tau)$$