Strict vs Weak Stationarity
Hierarchy
$$\text{Strict} \;\Longrightarrow\; \text{Weak} \quad (\text{if } E(X_t^2) < \infty)$$$$\text{Weak} \;\not\Longrightarrow\; \text{Strict}$$The Gaussian Bridge (★ 记住)
$$\text{Weak stationarity} + \text{Gaussian process} \;\Longrightarrow\; \text{Strict stationarity}$$| Condition | Strict → Weak? | Weak → Strict? |
|---|---|---|
| General | ✓ (finite 2nd moment) | ✗ |
| Gaussian | ✓ | ✓ |