Variational series for the scheme of summing independent variables

Let $\xi_1,\dots,\xi_N$ be $N$ independent observations of a random variable $\xi$ and $\xi_{(m)}$ be the $m$th order statistic of this sample. We study the asymptotic behaviour of $\xi_{(m)}$ and $\xi_{(N-m+1)}$ when the distribution of $\xi$ is a convolution of $n$ identical distributions and $n,N\to\infty$.