From Wikipedia

the family of complex normal distributions characterizes complex random variables whose real and imaginary parts are jointly normal,1 i.e., normally distributed and independent.

The definition in the same article is

Suppose $X$ and $Y$ are random vectors in $\mathbb{R}^k$ such that $vec[X Y]$ is a $2k$-dimensional normal random vector. Then we say that the complex random vector $$ Z = X + iY \, $$ has the complex normal distribution.

Does "independent" mean independence between random variables $X$ and $Y$, or linear independence between vectors $X$ and $Y$ in vector space $\mathbb{R}^k$ ?

If the former, why doesn't the definition require $X$ and $Y$ to be independent?



It means the random variables are statistically independent.

  • $\begingroup$ If it means s the random variables are statistically independent, why doesn't the definition require X and Y to be independent? $\endgroup$ – Tim Mar 8 '14 at 15:35
  • $\begingroup$ The page considers several different cases. In some cases, X and Y are independent, and in other cases they are not. I would say that the first sentence on that page is misleading. $\endgroup$ – Tom Minka Mar 8 '14 at 15:46
  • $\begingroup$ " the first sentence is misleading", do you mean the definition of complex Gaussian distribution shouldn't require independence between X and Y? $\endgroup$ – Tim Mar 8 '14 at 15:48
  • $\begingroup$ Yes. Looking at the history of that page, the comment about independence was added well after the rest of the page was written. $\endgroup$ – Tom Minka Mar 10 '14 at 12:08

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