Cross-Correlation of two signals each with independent complex Gaussian noise
I don't understand why your complex vectors have different dimensions but there is an answer for a scaler case, 

from:

Leo A. Goodman. On the exact variance of products. Journal of the American Statistical Association, 55(292):708–713, 1960.

Some definitions, 

$$\begin{array}{c}
m_{x1}= E\{ x_1 \}\\
var(x_1) = E\{ |x_1|^2\} - |m_{x1}|^2 \\
m_{x2}= E\{ x_2 \} \\
var(x_2) = E\{ |x_2|^2\} - |m_{x2}|^2 
\end{array} $$
for $x_1$ and $x_2$ independent,

$$ var(x_1 x_2^{*}) = var(x_1)|m_{x2}|^2 +var(x_2^*)|m_{x1}|^2 + var(x_2^*)var(x_1) $$

You don't specify the distribution of your signals.
assuming deterministic signals

N. O’Donoughue and J. M. F. Moura. On the product of independent complex gaussians. IEEE Transactions on Signal Processing, 60(3):1050– 1063, March 2012.