If I have $\mathbf x\_n=[x\_0, x\_1,... ,x\_K]^T$ and $\mathbf y\_n=[y\_0, y\_2, ..., y\_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$. 


What is the distribution of the following inner product: 
$$\sum\_{n=0}^N \mathbf x\_n^H\mathbf x\_n$$
Secondly, what is the distribution of
$$\sum\_{n=0}^N \mathbf x\_n^H\mathbf y\_n$$ 
If the answer is Gamma distribution, what are the parameters of this Gamma distribution? 
Note that each element in both vectors is complex, random, and independent of the other elements. 


Thank you in advance. 



The distribution of the sum of inner products of two independent complex normal vectors
$\require{amsmath}
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I understand the setting as follows: For $n=0,\dots,N$, let $\X\_n:=(X\_{n,0},\dots,X\_{n,k})$ and $\Y\_n:=(Y\_{n,0},\dots,Y\_{n,k})$, where all the $X\_{n,j}$'s and $Y\_{n,j}$'s are iid $\mathcal C\mathcal N(\mathbf 0,\si^2\mathbf I\_k)$. 


The first problem is to find the distribution of
$$S\_1:=\sum\_{n=0}^N\X\_n^H\X\_n=\sum\_{n=0}^N\sum\_{j=0}^k|X\_{n,j}|^2. 
$$


As for your second question, the distribution of 
For each pair $(n,j)$, we have $\frac2{\si^2}|X\_{n,j}|^2\sim\chi^2\_2$. So, $\frac2{\si^2}\,S\_1\sim\chi^2\_{2(k+1)(N+1)}=\text{Gamma}((k+1)(N+1),2)$. So, 
$$\sum\_{n=0}^N\X\_n^H\X\_n\sim\text{Gamma}((k+1)(N+1),\si^2),
$$
as you expected. If 
$$M:=(k+1)(N+1)
$$
is large, then, by the central limit theorem, 
$$\text{Gamma}((k+1)(N+1),\si^2)\approx\N(M\si^2,M\si^4). 
$$


As for the distribution of 
$$S\_2:=\sum\_{n=0}^N\X\_n^H\Y\_n=\sum\_{n=0}^N\sum\_{j=0}^k \overline{X\_{n,j}}\,Y\_{n,j}, 
$$
it is the $(k+1)(N+1)$-fold convolution of the distribution of the complex-valued random variable $\overline X\,Y=X\_1Y\_1+X\_2Y\_2+i(X\_1Y\_2-X\_2Y\_1)$, where $X:=X\_1+iX\_2$, $Y:=Y\_1+iY\_2$, and $X\_1,X\_2,Y\_1,Y\_2$ are iid $\mathcal N(0,\si^2/2)$. 


In turn, the distribution of $\overline X\,Y$ can be obtained by the transformation-of-distributions technique (i.e., change of variables in a multifold integral; see e.g. [Lecture 2](https://faculty.math.illinois.edu/~r-ash/Stat/StatLec1-5.pdf)) and is likely unremarkable. Mathematica worked several hours on getting the distribution of $\overline X\,Y$ and came up with nothing. 


However, the mean and covariance matrix of the joint distribution of $(\Re(\overline X\,Y),\Im(\overline X\,Y))=(X\_1Y\_1+X\_2Y\_2,X\_1Y\_2-X\_2Y\_1)$ are $[0,0]^T$ and $\si^4 I\_2/2$. So, if, again, $M$ is large, then, by the multivariate (here bivariate) central limit theorem, the joint distribution of $(\Re S\_2,\Im S\_2)$ is approximately the bivariate normal distribution
$$\N(0,0,M\si^4/2,M\si^4/2,0),
$$
with zero means, both variances equal $M\si^4/2$, and zero correlation. 
That is, the real and imaginary parts of $\overline X\,Y$ are (i) zero-mean, (ii) each with variance $M\si^4/2$, (iii) jointly asymptotically normal, and (iv) asymptotically independent. 


