Is there any hint on how to calculate, or at least attain a upper bound on, the following integral?
$\int_0^\infty\left[\frac{\Gamma(n,x)}{\Gamma(n)}\right]^Kdx$.
This is in fact the expectation of the minimum of $K$ i.i.d. Gamma random variables, each with shape $n$ and scale $1$. In my problem, $n$ is integer, hence
$\frac{\Gamma(n,x)}{\Gamma(n)} = e^{-x}\sum_{m=0}^{n-1}\frac{x^m}{m!}$.
I would like to see the asymptotic behavior of this expectation when $K\to\infty$ and $n$ remains fixed. From numerical simulation, it seems vanishing as $1/K$. I want to justify this rigorously. Any other method without dealing with this integral would be appreciated.
Thank you.