Cross-Correlation of two signals each with independent complex Gaussian noise The nearest question I can find is here, but it does not have an answer. Say I want to correlate the following two noisy channels: $x_1 = s_1 + n_1$ and $x_2 = s_2 + n_2$, where $s_1$ and $s_2$ are arbitrary signals and $n_1$ and $n_2$ are independent complex Gaussian noise (identically distributed for starters). Further, let either $M \lt N$ or $M \ll N$, with $x_1 \in \mathbb{C}^M$ and $x_2 \in \mathbb{C}^N$. The cross-correlation of the two channels yields

$$
x_1 \star x_2 = s_1 \star s_2 + s_1 \star n_2 + n_1 \star s_2 + n_1 \star n_2.
$$

Perhaps I am searching with the wrong keywords, but I am having trouble finding information about $n_1 \star n_2$. What kind of distribution does it have? How much power does it contribute to the output?
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.