Independence vs Uncorrelatedness
Definitions
- Independent: $P(X \leq x, Y \leq y) = P(X \leq x)\,P(Y \leq y)$ for all $x, y$. The full joint distribution factors.
- Uncorrelated: $\text{Cov}(X, Y) = 0$. Only a statement about second moments — no linear relationship.
Key Relationship
$$\text{Independent} \;\Longrightarrow\; \text{Uncorrelated}$$$$\text{Uncorrelated} \;\not\Longrightarrow\; \text{Independent}$$The Gaussian Exception
If $(X, Y)$ are jointly Gaussian:
$$\text{Uncorrelated} \;\Longleftrightarrow\; \text{Independent}$$A Gaussian distribution is fully characterized by its first two moments. This is why for Gaussian processes, weak stationarity ⟺ strict stationarity, and uncorrelated noise becomes independent noise.
Why This Matters in STA457
- White Noise (WN) requires only uncorrelatedness → weakly stationary, may not be strictly stationary.
- IID Noise requires independence → always strictly stationary.
- q-correlated vs q-dependent: MA(q) is always q-correlated, but only q-dependent when noise is Gaussian.