We define a process $\chi_k^n=\sum _{i=1}^k x_i^n$ where x_i are iid gaussian processes. I try to find the distribution of $\chi_k^n$. If k=1 then we get $f(x^n=y)=\frac1n y^{\frac{1-n}{n}}\exp(-y^{2/n}/2)$ and then try to do convolution but then to say something about the integral $$C_{n}\int_{-\infty}^{\infty}\frac1n (l^2-h^2)^{\frac{1-n}{n}}\exp(-(l+h)^{2/n}/2)\exp(-(l+h)^{2/n}/2)\exp(-(l-h)^{2/n}/2)dh $$ What can I do n>2????????

Can I say somthing about the density, formula or upper bound?

Thanks.