I have an exact CDF ($F_{X_i}(x)$) and PDF ($f_{X_i}(x)$) of a random variable $X_i\geq 0$ which includes special functions. Since I needed an approximation around $x=0$, I therefore derived an approximation as $$F_{X_i}(x)\approx-a x \log(a x)$$ $$f_{X_i}(x)\approx-a (\log(a x)+1)$$ where $a>0$, and this CDF is a tight one.
However, now I need the CDF of $$Y=\sum_{i=1}^{N}X_i$$ where $X_i$s are i.i.d.
I tried to bound this by $\left( F_{X_i}(x)\right)^N$ or $\left( F_{X_i}(x/N)\right)^N$. I found that these are too loose even for small $N$.
If anyone has a good idea to derive $F_Y(y)$ more accurately at small $x$, please share with me.