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

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 $$t$$??

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

• Can you give a reference for that claim? All sources I am aware of state the condition for all $h$, as you do on the r.h.s. of the presumed implication. Jun 17, 2020 at 14:07
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.