Hi everyone,

I am a little bit confused between the two terms stationarity and weak dependence. Can someone explain to me the difference between these two concepts?

From my point of view (covariance) stationarity means that

- $\displaystyle E(x_t)=c, \quad \text{where c is a constant.}$
- $\displaystyle var(x_t)=k, \quad \text{where k is a constant.}$
- $\displaystyle cov(x_t,x_{t+h})=p_h, \quad \text{where } t, h \geq 1 \text{ and } p_h \text{ depends on } h, \text{ and not on } t.$

Ok, so far I can follow. The definition of weak dependence:

- $\displaystyle corr(x_t,x_{t+h})\to 0 \text{ for } h \to \infty$

I also understand that. But, most textbooks state that the correlation has to approach zero sufficiently quickly. What does "sufficiently" mean?

Now here is my confusion. When a time series is not weakly dependent, i.e. the $\displaystyle corr(x_t,x_{t+h})$ does not approach zero when h approaches infinity, does that mean that the time series is also not stationary? If so, what is the difference between these two. Can there be time series that are stationary but not weakly dependent? Can there be time series that are not stationary but weakly dependent? Can there be time series that are stationary and weakly dependent?

What is a good way to test for stationarity? Dickey/Fuller does the job in my opinion. But what about weak dependence? How do we test for that?

I am sorry, I couldn't come up with one single question, because I don't really get the whole concept yet. So I'll be thankful for any comments that clarify the difference between stationarity and weak dependece. Thanks a lot.