# Would it be possible to have a time series which has zero-mean but is not stationary?

E.g. $y_t= A_1y_{t−1} + u_t, u_t ∼ (0, \Sigma_u)$

Would it be possible to let the time series to have zero-mean but is not stationary?

• E.g. mean stays constant, but variance changes over time.
– Tim
Jun 1 '18 at 21:17
• It seems that in some text book (e.g. afriheritage.org/TTT/…), "zero-mean" seems to mean that the intercept is 0. I want to be sure about that. Jun 2 '18 at 4:35

$$y_t = y_{t-1} + u_t, \;\;\; u_t \sim WN(\sigma^2)$$
Since then $y_t = \sum_{i=1}^t u_i$ it follows that
$$E(y_t)=0, \text{Var}(y_t) = t\sigma^2$$
Then $y_t$ is first-order stationary (constant mean), but not second-order (or weakly) stationary.