ACF of a stationary time series

Two questions:

1. Why for a stationary time series the ACF will drop to zero relatively quickly? Stationarity is a form of dependence and the ACF (and the PACF) measure the dependence between two r.v's, it's seems a contradiction to me (of course I'm wrong, but I don't understand why).

2. We can use Ljung-box test to test stationarity. Stationarity is a form of dependence. If the series is stationary I'll refuse the null hypothesis, that is independence. Is it correct?

• Stationarity is a form of dependence? On what? – Wayne May 8 '17 at 20:55
• @Wayne Between random variables of the process – zar May 8 '17 at 21:09

1. It depends on what you mean by "drop to zero relatively quickly". For a stationary AR(1), $y_t = \rho y_{t-1} + \epsilon_t$, $|\rho| < 1$, the ACF never goes to zero, but rather decays as $\rho^k$. In contrast, for an MA(1), $y_t = \alpha \epsilon_{t-1} + \epsilon_t$, the $E[y_t y_{t+k}] = 0$ for $k > 1$, so it drop to zero very quickly.