Weakly Stationary

Definition

Let $\gamma$ : Function of the lag $k$ and is independent of $t$.

A stochastic process $\{Y_t\}$ is weakly stationary

If it satisfies two conditions:

1. $E[Y_t]=\mu$ (constant mean)
2. $\mathrm{Cov}(Y_t,Y_{t-k})={\gamma }_k$ (covariance independent of $t$)

Weakly Stationary Process has Constant Variance

$\{Y_t\}$ is weakly stationary, then ${\gamma }_0=\text{Cov}(Y_t,Y_{t-0})=\text{Var}(Y_t)$ is constant, independent of $t$.

Relationship with Strict Stationarity

Relationship with Weak Stationarity

Let $\{Y_t\}$ be strictly stationary process.

If $\text{Var}(Y_t)$ is finite for all $t$

The $\{Y_t\}$ is also weakly stationary

Converse is Not True

Weak stationarity does NOT imply strict stationarity. A process can have constant mean and covariance without having identical distributions at different times.

Illustration