On the limit distributions for middle order statistics in a scheme of series

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. Asymptotic behaviour of $\xi_{(m)}$ is studied when the distribution of $\xi$ is the convolution of $n$ identical distributions and $n\to\infty$, $m\to\infty$ $N-m\to\infty$ together with $N\to\infty$.