Stationary time series

Strongly stationary

If the joint distribution of $X(t_1), \ldots, X(t_n)$is the same as the joint distribution of $X(t_1+\tau), \ldots, X(t_n+\tau)$.

Weakly stationary

Also called second-order stationary. If the conditions below are satisfied:

(Note: $\text{Cov}(X,Y) = \text{E} [(X-\mu_x)(Y-\mu_y)]$.) No assumptions need to be made for higher moments.

Your data sample is $[x(t_1), x(t_2), \cdots, x(t_n)]$, a realization of the joint distribution $[X(t_1), \ldots, X(t_n)]$.

This stationary condition is necessary for applying the methods described in this section, since it is required by most of the detailed derivations, especially the Wiener-Khintchine theorem (see 2.4).

Is your time series (weakly) stationary?

• Detrend the data first.
• Think about if the conditions are unchanged for the entire interval when the signals were recorded.
• The change of perturbation amplitude with time does not necessarily mean that your time series is not (weakly) stationary.