Consider two random variables $X_t$ and $X_{t+\tau}$; where both come from random variable $X$ but are lagged by $\tau$. Assume that both are mean-free. If we want calculate covariance matrix of them, then we do:

$$\Sigma = X_t^T X_{t+\tau}$$ if we perform eigendecomposition

$$A\Sigma = \Sigma\Lambda$$

Then if the time lagged covariance is properly converged, all the eigenvalues are going to be positive, non-complex ?

  • $\begingroup$ What do you mean by "properly converged"? $\endgroup$
    – mhdadk
    Nov 25 at 19:07
  • $\begingroup$ @mhdadk There's enough samples to calculate it, so that it does not change much if we add more samples $\endgroup$ Nov 25 at 20:12
  • $\begingroup$ One could check for diagonal dominance, see math.stackexchange.com/questions/87528/…. By intuition, if autocorrelation $<X_{t}, X_{t+\tau}>$ is much higher than the rest, it is probable that $\Sigma$ is positive definite. $\endgroup$ Nov 25 at 23:36

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