$X_t=_1+\dots+_$ where $\{_\}_{\geq 1}$ is a white noise having the following properties :
- $_ \sim \left(0,^2\right)$ $\forall t\geq 1$
- $\gamma_(h)=0\: ∀ h≠0$
- $_$ ⫫$_{+ℎ}\: ∀ ,∀ ℎ≠0$.
Show that the above equation is not weak sense stationary in detailed form.
I do know somewhat what to do but I don't understand the correlation part...
Observe that $X_t$ is square integrable for all $t$ and that $\mathbb E\left[X_t\right]=0$ hence in order to disprove the weak stationarity, it suffices to prove that the sequence $\left(\mathbb E\left[X_tX_{t+1}\right]\right)_{t\geqslant 1}$ is not constant. To see this, write $X_tX_{t+1}=X_t\left(X_t+Z_{t+1}\right)=X_t^2+X_tZ_{t+1}$. Use the fact that $Z_{t+1}$ is centered and independent of (hence uncorrelated to) $Z_i$ for all $i\in\{1,\dots,t\}$ to get that $\mathbb E\left[X_tZ_{t+1}\right]$. And the expectation of the remaining term $X_t^2$ depends heavily on $t$. It suffices to do the computations for $t=1$ and $t=2$.