# Thread: Is an Ergodic process weakly stationary?

1. ## Is an Ergodic process weakly stationary?

I have this true false question on prior exams I am preparing for, is an ergodic process weakly stationary?

As far as I understand it, an ergodic process means that any two "Zt's" or variables are asymptotically independent so they are a bit similar but not much. A weakly stationary process means that the covariance between the two Zt's depens only on the distance in the time series, so the further away they are in time the more their covariance tends towards zero.

It seems the concepts are similar, the variance between the variables asymptotically fading away in the time series.

But can we say "An ergodic process IS weakly stationary" and why?

Thanks!
TONY S

2. ## Re: Is an Ergodic process weakly stationary?

It depends which definition of ergodicity you have at your disposal. For example an independent sequence will form an ergodic sequence using the Kolmogorov 0-1 law (it means that each invariant set has measure 0 or 1). But if we take for example $\displaystyle Y_j:= jX_j$, where $\displaystyle (X_j)$ is iid, we obtain an ergodic process which is not weakly stationary.