Let $\Sigma $ be the covariance matrix of a multivariate normal distribution.

Thereotically, I guess it does not have to to strictly positive definition, i.e., $\det (\Sigma )=0$ is possible.

In this case, the density function does not exist, how can I generate random numbers(vectors) from such multivariate normal distribution?

  • 2
    $\begingroup$ To simulate $X$ with covariance $\Sigma$ or size $n$ and rank $k$, choose $L$ of size $n\times k$ such that $\Sigma=LL^T$, simulate $U$ standard normal of size $k$ and use $X=LU$. $\endgroup$ – Did Sep 2 '17 at 19:25
  • $\begingroup$ @MANMAID "so I am not sure, if this works or not" Or not. $\endgroup$ – Did Sep 2 '17 at 19:26

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