pr.probability - The density of x_1^n+x_2^n where x_i are Gaussian

Firstly you forgot to multiply the
density $f(x^n=y)$ by $1/sqrt{2pi}$. I think if you
obtained the density of the random variable $X_{1,2}=X_1^n+X_2^n$ by
the convolution method, the problem no more posed , because for
$X_1^n+X_2^n+X_3^n=X^n_{1,2}+X_3^n=X^n_{1,2,3}$, and you have the
density of $X^n_{1,2}$, the density of $X^n_{3}$, you can calculate
there convolution, i.e the density of $X^n_{1,2,3}$. If the calculation is
very difficult with the convolution (I think) you can
use the characteristic function. You calculate the function
characteristic of the variable $X_1^n$ that one noted
$psi_{X_1^n}(t)$. As the two variables $X_1^n$ and $X_2^n$ are
i.i.d, then $psi_{X_1^n+X_2^n}(t)=psi_{X_1^n}(t)cdot
psi_{X_2^n}(t)=(psi_{X_1^n}(t))^2$ and so on for
variable $X_1^n+X_2^n+...+X_k^n$ we will have
$psi_{X_1^n+X_2^n+ldots +X_k^n}(t)=(psi_{X_1^n}(t))^k$. Just well calculate $psi_{X_1^n}(t)$.