Let $\bf g$ is an $N$-by-$1$ complex Gaussian random vector, whose distribution $CN(0,{\bf I})$. Consider an $N$-by-$M$ complex matrix ${\bf A}$, whose columns are orthogonal to one another and have unit norm each. Denote the $m$th column vector of ${\bf A}$ by ${\bf a}_m$.
My question is what is the distribution of the following scalar value $x$ when $M$ is very large (or asymptotically infinity):
$x=\frac{1}{M}\sum_{m=1}^{m=M} {\bf g}^H {\bf a}_m$
Start by separating the real and imaginary parts of $x$. Then consider that the sum of gaussians is gaussian, so both the real and imaginary parts of $x$ will be gaussian. The mean of $x$ is $0$ by linearity and the variance should be straightforward since all the components of $\boldsymbol{g}$ are independent.