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 three conditions:

1. $E[Y_t]=\mu$ (constant mean)
2. $\mathrm{Var}(Y_t)<\infty ,\forall t$ (finite variance)
3. $\mathrm{Cov}(Y_t,Y_{t-k})={\gamma }_k$ (cov independent of $t$)

Relationship with Strict Stationarity

If a stochastic process $\{Y_t\}$ is strictly stationary and $\mathrm{Var}(Y_t)$ is finite for all $t$, then $\{Y_t\}$ is also weakly stationary.