IID vs White Noise
Hierarchy
$$\text{IID}(0, \sigma^2) \;\subset\; \text{WN}(0, \sigma^2)$$IID noise is a special case of white noise (independence is stronger than uncorrelatedness).
Comparison
| Property | IID$(0, \sigma^2)$ | WN$(0, \sigma^2)$ |
|---|---|---|
| Zero mean | ✓ | ✓ |
| Constant variance | ✓ | ✓ |
| Uncorrelated | ✓ | ✓ |
| Independent | ✓ | Not necessarily |
| Weakly stationary | ✓ (if $\sigma^2 < \infty$) | ✓ |
| Strictly stationary | ✓ | Not necessarily |
Key Exam Statements (True/False)
- “White noise is a stationary process” → True (weakly)
- “If a process is stationary, then it is white noise” → False (e.g., MA(1) is stationary but not WN)
- “IID noise with finite variance is a special case of white noise” → True
- “A weakly stationary Gaussian process is strictly stationary” → True