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New limiting distributions of maxima of independent random variables. (English) Zbl 1042.60032

Let \(\{X_n\}\) be a sequence of independent and, in general, no identically distributed random variables, and let \(\{k_n\}\) be an integer sequence satisfying \(k_{n+1}\geq k_n\geq 1\) and \(\lim_{n\to\infty}k_n=\infty\). Under some uniformity assumption, the author characterizes the class of all non-degenerate distribution functions \(G(x)\) in \(\lim_{n\to\infty}P\left(M_{k_n}\leq x/a_n+b_n\right)=G(x)\, ,\) where \(M_{k_n}=\max\{X_1,\ldots,X_{k_n}\}\) and \(\{a_n\},\,a_n>0\), and \(\{b_n\}\) are real sequences. This class contains the log-concave and max-semistable distributions.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60E05 Probability distributions: general theory
60F05 Central limit and other weak theorems
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