On Limit Theorems for the Distribution of the Maximal Element in a Sequence of Random Variables

Abstract

Theory of Probability &Its Applications, Volume 69, Issue 2, Page 186-204, August 2024.
We study the distribution of the maximal element $\overline{\xi}_n$ of a sequence of independent random variables $\xi_1,\dots,\xi_n$ and not only for them. The presented approach is more transparent (in our opinion) than the one used before. We consider four classes of distributions with right-unbounded supports and find limit theorems (in an explicit form) of the distribution of $\overline{\xi}_n$ for them. Earlier, only two classes of right-unbounded distributions were considered, and it was assumed a priori that the normalization of $\overline{\xi}_n$ is linear; in addition, the components of the normalization (in their explicit form) were unknown. For the two new classes, the required normalization turns our to be nonlinear. Results of this kind are also obtained for four classes of distributions with right-bounded support, which are analogues of the above four right-unbounded distributions (earlier, only the class of distributions with right-bounded support was considered). Some extensions of these results are obtained.