Is it enough for the ACF to be defined at $1$ for it to be defined at every other natural

Question

I've read that a time series should be a weakly stationary for the Autocorrelation function (ACF) to make sense. The definition for weakly stationary series that I have is that all the observations should have the same mean and variance and that $\mathrm{Cov}(x_t,x_{t-1})$ should be same for all $t$. But ACF for a natural number $h$ is defined to be the correlation of $x_t$ and $x_{t-h}$. So, for it to be defined as a function, we should have $\mathrm{Cov}(x_t,x_{t-h})$ independent of $t$. How is this implied by $\mathrm{Cov}(x_t,x_{t-1})$ is independent of $t$??

$\mathrm{Cov}(x_t,x_{t-1})\text{ is independent of }t\implies\mathrm{Cov}(x_t,x_{t-h}) \text{ is independent of } t\,\forall h\in\Bbb N$??

Please help.

Answer 1

Consider an AR(2) process $$ Y_t=\phi_{2}Y_{t-2}+\epsilon_t, $$ where $\phi_{2}=\phi_{2,1}$ for $t=-\infty,\ldots,[sT]$ and $\phi_{2}=\phi_{2,2}$ for $t=[sT]+1,\ldots,\infty$. Hence, the process is not stationary, because the properties of the process change after $[sT]$. However, before and after the break, $\phi_1$, the coefficient on the first lag, is zero.

By properties of AR(2) processes, the first autocorelation of this process is $$ \rho_1=\frac{\phi_1}{1-\phi_2}=0, $$ which is evidently independent of $t$. The second autocorrelation, however, is now given by $$ \rho_2=\phi_2, $$ which is not stable in view of the structural change in the process.