White Noise Process
White noise is the simplest stationary process and serves as the building block for all time series models.
1. Definition
A stochastic process $\{Z_t\}$ is a white noise process, denoted $\text{WN}(0, \sigma^2)$, if it is a sequence of uncorrelated random variables with:
- Zero mean: $\mathbb{E}[Z_t] = 0$
- Constant variance: $\text{Var}(Z_t) = \sigma^2$
- Autocovariance: $\gamma(h) = \begin{cases} \sigma^2 & h = 0 \\ 0 & h \neq 0 \end{cases}$
2. Stationarity
- White noise is always weakly stationary
- White noise may not be strictly stationary (strict stationarity requires identical joint distributions, not just uncorrelatedness)
- i.i.d. noise with finite variance is a special case of white noise that is also strictly stationary
- Uncorrelatedness here is strictly weaker than independence
3. True/False Exam Questions
| Statement | Answer |
|---|---|
| White noise is a stationary process | TRUE |
| If a process is stationary, then it is white noise | FALSE |
| i.i.d. noise with finite variance is a special case of white noise | TRUE |
| A weakly stationary Gaussian process is strictly stationary | TRUE |
| For any stationary $\{X_t\}$, $\text{Cov}(X_t, X_{t+h}) = 0$ for $h \neq 0$ | FALSE (only true for WN) |