I have read that the definition of weak stationary is :

$ Mean(t) = mean(t + \tau)\\ Cov(t_1,t_2) = cov(t_1-t_2,0)\\ E[|x(t)|^2] < \infty $

In this definition of weakly stationary, can the variance change as a function of time as long as the covariance is a constant function of $\Delta t$?

I have also read the definition of covariance stationary is: $$\exists \mu : E(x_n)=\mu \,\forall \, n>0\\ \forall \, j \geq 0 \,\exists \,\gamma_j : cov(x_n,x_{n-j})=\gamma_j \, \forall \, n > j$$

Are these definitions identical?

I have also read that variance must be a constant function of time if data is stationary or weakly stationary, but these definitions mention covariance and not variance. Does variance have to be constant for data to be stationary, or not? Is covariance always stationary if variance is constant?


1 Answer 1


The two definitions are nearly the same, except that the second concerns the process after $t=0$. And, in the first one, the notation is abused. It should be $$\begin{align} & E[x(t)]=E[x(t+\tau)]\\ &\operatorname{cov}(x(t_1),x(t_2))=\operatorname{cov}(x(t_1-t_2),x(0))\end{align}$$

The variance is a special case of the covariance where $t_1=t_2$, i.e. $$\operatorname{cov}(x(t_1),x(t_1))=\operatorname{var}(x(t_1))=\operatorname{var}(x(0))$$ which means variance is also constant. When variance is constant, it doesn't mean that the process is covariance stationary; but the other way is true as you can see.

  • $\begingroup$ Thanks! So weak stationary and covariance stationary are the same? $\endgroup$
    – Frank
    Jun 20, 2019 at 0:12
  • 2
    $\begingroup$ Yes, check it here: wikiwand.com/en/Stationary_process#/… $\endgroup$
    – gunes
    Jun 20, 2019 at 0:14
  • 1
    $\begingroup$ @Frank No, weak stationarity and covariance stationarity are very nearly the same. The difference is that a more general definition of covariance stationarity than the one given to you does not insist that all the random variables have the same mean, as weak stationarity does, and so the class of covariance-stationary processes is a superset of the class of weakly stationary processes. Covariance-stationary processes that are not weakly stationary have mean functions that vary with time; see the vast literature on trend-stationary processes for examples. $\endgroup$ Jun 20, 2019 at 3:11
  • $\begingroup$ In your last sentence, do you intentionally have the notion of covariance (rather than the process itself) being stationary? $\endgroup$ Jun 20, 2019 at 9:22
  • $\begingroup$ I meant the process being covariance stationary. Edited. $\endgroup$
    – gunes
    Jun 20, 2019 at 9:27

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