0

I have a problem proving the following statement: if $A$ and $B$ are positive definite $n\times n$ matrices, then there exist independent random vectors $X$, $Y$ such that $X\sim N(0,A)$ and $Y\sim N(0,B)$. I know that if $A$ is positive definite, then for any $m$ there exist a random vector $X$ such that $X\sim N(m,S)$, but how to make $X$ and $Y$ independent?

develarist
  • 3,009
  • 8
  • 31
Maria
  • 1
  • 1
  • 1
    Construct a joint distribution with a block-diagonal covariance matrix, the blocks being $A$ and $B$. For the Normal distribution, this suffices to make $X$ and $Y$ independent, since the covariance between them will be zero and (for the Normal) this implies independence. – jbowman Sep 23 '20 at 19:08
  • This is not a special property of Normal variables: there is a generic procedure, described in detail in my post at https://stats.stackexchange.com/a/352897/919, for constructing independent random variables out of given distributions. – whuber Sep 23 '20 at 19:47

0 Answers0