TITLE: Variance of a sum of complex independent random variables
QUESTION [8 upvotes]: $\newcommand{\Var}{\operatorname{Var}}$Consider the zero mean independent complex random variables $X_1,\dots,X_n$ and the complex constants $a_1,\dots,a_n$. Does the formula for real valued independent random variables carry over to complex case as: $$\Var[a_1X_1+\dots +a_nX_n] = | a_1 | ^2\Var(X_1)+\dots+ | a_n |^2\Var(X_n)$$

REPLY [9 votes]: $\newcommand{\var}{\operatorname{var}}$
The variance is often defined as
$$
\var(X) = \operatorname E\left((X-\mu)\overline{(X-\mu)}\,\right)
$$
where $\overline{c}$ is the complex conjugate of $c$.
It follows that
$$
\var(aX) = a\overline{a}\var(X) = |a|^2\var(X).
$$
With real numbers $|a|^2$ is the same as $a^2$; with complex numbers they are not equal except when the complex number involved is real.
The statement about sums is true and the proof is the same as with real numbers.