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Confused about the definition of a weakly stationary stochastic process. Wikipedia and this Stack question have it that a process is weakly stationary if it has

  • A constant mean $m_X(t)$
  • Covariance $\operatorname{Cov}(X_t, X_s)$ depending only on $t-s$
  • Finite (and constant) auto-covariances $\operatorname{Var}(X_t)$

Then on the Wikipedia page for the Ornstein-Uhlenbeck process, it describes the process as "stationary" (doesn't say weak or strict, but evidently weak) but then gives the mean and the covariance, and they are not as above. For instance,

$$\operatorname{Cov}(X_t, X_s) \propto \left(e^{- \theta |t-s|} - e^{\theta(t + s)} \right)$$

This evidently does not depend only on $t-s$. What gives?

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Eric Auld
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1 Answers1

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In order to be stationary, the O-U process needs to begin with a Gaussian initial condition. This discussion is about that.

https://stats.stackexchange.com/questions/365881/dealing-with-different-definitions-of-the-ornstein-uhlenbeck-process

If it starts with a constant initial condition, then you get that covariance that you mention. When the wikipedia article calls it stationary, it probably could have been clarified at that point.

Hope that helps. Greg

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