# Characteristic function of sum of squares of Gaussian distributed variables

Let $$X_1, X_2, \ldots X_n$$ be independent Gaussian random variables with $$N(0,1)$$ distribution. Find the density and the characteristic function of $$X_1^2 + X_2^2 + \cdots + X_n^2$$

I am preparing for exams and I am stuck with this particular problem. I have seen the derivation for the characteristic function of a standard normal random variable. But I tried calculating the integral for the square and I cannot get it. Once I do that though, I think I can just raise my answer to the nth power to account for the sum. Can someone please help me with this? I have been stuck and not sure for many hours.

Let $$Y=\sum_{k=1}^n X_k^2$$. The char. function of $$Y$$ is $$\phi(t)=E(e^{itY})=\prod_{k=1}^nE(e^{itX_k^2})$$, because of independence.
$$E(e^{itX_k^2})=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-\frac{x^2}{2}+itx^2}dx=g(t)$$. Therefore $$\phi(t)=g^n(t)$$.
I believe $$g(t)=\frac{1}{\sqrt{1-2it}}$$.
• Should it be $g(t) = \frac{1}{\sqrt{1 - 2it}}$? – hom Dec 3 at 1:36
• Yes - I got mixed up in the final step to get $g(t)$.. I corrected the answer. Getting the density function seems tough. Fourier inversion is correct, but it looks very difficult. – herb steinberg Dec 3 at 4:49
• @hom I believe that the density function might be derivable by diirect integration. I did the first three $(Y_n=\sum_{k=1}^n X_k^2)$ and $f_n(y)$ are the density functions for$Y_n$ I got $f_1(y)=\frac{e^{-y/2}}{\sqrt{2\pi y}}$ , $f_2(y)=\frac{e^{-y/2}}{2}$, and $f_3(y)=e^{-y/2}\sqrt{\frac{y}{2\pi}}$ – herb steinberg Dec 5 at 1:39
• @hom Since $f_2(y)$ is so simple, you can the rest of list by doing the odd and even numbered members as two lists by repeated convolutions using $f_2(y)$. – herb steinberg Dec 5 at 3:38