A Generalization of the Mejzler--de Haan Theorem

Let $(k_n)$ be a sequence of positive integers such that $k_n\to \infty$ as $n\to\infty$. Let $X^\ast_{n1}, \ldots,X^\ast_{nk_n}$, $n\in{\bf N}$, be a double array of random variables such that for each $n$ the random variables $X^\ast_{n1},\ldots, X^\ast_{nk_n}$ are independent with a common distribution function $F_n$, and let us denote $M^\ast_n=\max\{X^\ast_{n1},\dots,X^\ast_{nk_n}\}$. We consider an example of double array random variables connected with a certain combinatorial waiting time problem (including both dependent and independent cases), where $k_n=n$ for all $n$ and the limiting distribution function for $M^\ast_n$ is $\Lambda(x)=\exp(-e^{-x})$, although none of the distribution functions $F_n$ belongs to the domain of attraction $D(\Lambda)$. We also generalize the Mejzler--de Haan theorem and give the necessary and sufficient conditions for the sequence $(F_n)$ under which there exist sequences $a_n>0$ and $b_n\in {\bf R}$, $n\in{\bf N}$, such that $F_n^{k_n}(a_nx+b_n)\to\exp(-e^{-x})$ as $n\to\infty$ for every real $x$.