Stationarity

Stationarity is the fundamental simplifying assumption in time series analysis: the statistical properties of the process do not change over time.

1. Strict Stationarity

A stochastic process $\{X_t, t \in T\}$ is strictly stationary if for any $n \geq 1$, any times $t_1, \dots, t_n$, and any lag $h$, the joint distribution of $(X_{t_1}, \dots, X_{t_n})$ is the same as that of $(X_{t_1+h}, \dots, X_{t_n+h})$.

That is:

$$\mathbb{P}(X_{t_1} \leq x_1, \dots, X_{t_n} \leq x_n) = \mathbb{P}(X_{t_1+h} \leq x_1, \dots, X_{t_n+h} \leq x_n)$$

2. Weak (Second-Order) Stationarity

A stochastic process $\{X_t\}$ is (weakly) stationary if:

  1. $\mathbb{E}(X_t) = \mu$ is constant (independent of $t$)
  2. $\text{Var}(X_t) = \sigma^2$ is constant (independent of $t$)
  3. $\text{Cov}(X_t, X_s) = \text{Cov}(X_{t+h}, X_{s+h})$ for all $t, s, h$

Condition 3 means the covariance depends only on the lag $h = |t - s|$, not on the time index itself.

3. Relationship

  • Strict stationarity $\Rightarrow$ weak stationarity (provided $\mathbb{E}(|X_t|) < \infty$)
  • Weak stationarity $\not\Rightarrow$ strict stationarity in general
  • Key exception: Weak stationarity + Gaussian process $\Rightarrow$ strict stationarity

This is because a Gaussian process is fully characterized by its mean and covariance structure.

4. Consequences of Weak Stationarity

  • Constant mean function: $\mu_X(t) = \mathbb{E}(X_t) = \mu$
  • Constant variance function: $\sigma_X^2(t) = \text{Var}(X_t) = \sigma^2$
  • Autocovariance Function (ACVF) depends only on lag: $\gamma_X(t, s) = \gamma_X(h)$ where $h = |t-s|$

5. Examples

  • A sequence of i.i.d. random variables is always strictly stationary
  • White Noise Process $\text{WN}(0, \sigma^2)$ is always weakly stationary, but may not be strictly stationary
  • A process with time-dependent mean $\mu(t) = t$ is not stationary