## 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

### Keywords:

extremes; log-semiconcave distributions
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