# Does mean reverting imply mean stationary?

If I have a time series that exhibits mean reverting properties, does it necessarily mean that the time series is mean stationary?

• What is mean stationary? A quick Google search brings up asymptotically mean stationary. – chrisaycock Nov 16 '12 at 1:57
• It is possible that he means covariance stationary. – John Nov 16 '12 at 14:46
• "exhibiting mean reverting properties" is pretty general in any case...it could be as weak as the power spectrum having small size for long wavelengths – Brian B Nov 16 '12 at 16:32
• Mean stationary means that $E(x_t) = E(x_s)$ for all $s$ and $t$. – Ryogi Nov 16 '12 at 17:34

## 1 Answer

As pointed out by Brian, the question is vague because generally mean reversion requires a well defined mean. Nevertheless, there are processes which are not mean stationary (mean is not homogenous across observations) for which a concept of mean exists. Let $\mu_t = E(x_t)$. In general you can have $\mu_t \neq \mu_s$ (i.e. violate mean stationarity) but have a well defined long run mean, i.e. the limit

$$\frac1n \sum_{t=1}^n \mu_t \to \bar \mu$$

exists. In such a situation, you can define a concept of mean reversion to the long run mean that applies to non mean stationary processes.