# The distribution of the sum of inner products of two independent complex normal vectors

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.

$$\require{amsmath} \require{graphicx} \newcommand{\X}{\mathbf X} \newcommand{\Y}{\mathbf Y} \newcommand{\N}{\mathcal N} \newcommand{\si}{\sigma}$$ 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) 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.

• Thanx losif Pinelis. When you say Gamma(X,Y), does that mean X is the mean and Y is the variance of the Gamma distribution or the shape and rate parameters? – Aymen Kareem Nov 21 at 17:01
• @AymenKareem : $\text{Gamma}(a,b)$ means the gamma distribution with shape parameter $a$ and scale parameter $b$, so that the mean is $ab$ and the variance is $ab^2$. – Iosif Pinelis Nov 21 at 22:22
• I have added normal approximations to the distributions of the two sums. – Iosif Pinelis Nov 22 at 1:23