Strict Stationarity
Definition
$\{X_t, t \in T\}$ is strictly stationary if for any $n \geq 1$, times $t_1, \ldots, t_n$, and lag $h$:
$$\Pr(X_{t_1} \leq x_1, \ldots, X_{t_n} \leq x_n) = \Pr(X_{t_1+h} \leq x_1, \ldots, X_{t_n+h} \leq x_n)$$The entire joint distribution is invariant under time shifts.
Consequences (if moments exist)
- Constant mean: $E(X_t) = \mu$
- Constant variance: $\text{Var}(X_t) = \sigma^2$
- ACVF depends only on lag
Example
i.i.d. random variables are always strictly stationary.