Let $x_1,\ldots,x_n$ be i.i.d. random variables with continuous and concave distribution function F.
It is known that for $t\geq 0$ $ P\left(\sum_{i=1}^nx_i\leq t\right)\leq\\ \left\{ \begin{array}{rcl} \frac{1}{n!}\sum_{j=0}^k(-1)^j {n \choose j}\left(nF\left(\frac tn\right)-j\right)^n, &nF^{-1}\left(\frac kn\right)\leq t
I am wondering if it is possible to bound the RHS of the inequality above (the sum). Is it possible by adding some assumptions on random variables $x_1,...,x_n$ to bound density function F?
Continuous distribution function $F$ with support $[0, \infty)$ is called concave, if $F(\lambda s+(1-\lambda t))\geq \lambda F(s)+(1-\lambda)F(t)$, for every $s,t\geq 0, 0\leq\lambda\leq 1$.
Any references and ideas would be very helpful.
Thank you.