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Limit theorems for extremes with random sample size. (English) Zbl 0916.60017

The main problem discussed in this interesting paper can be formulated as follows: For \( \varepsilon > 0 \) let \( \xi_{\varepsilon} : (0,\infty)\rightarrow (-\infty ,\infty) \) be an extremal process and let \( \nu_{\varepsilon} : (0,\infty) \rightarrow (0,\infty) \) be the associated stopping-index process (sometimes also called trading time (Mandelbrot), random time-change (Balkema), etc.). Without assuming independence, which conditions will guarantee the convergence of the composition \( \xi_{\varepsilon} \circ \nu_{\varepsilon} \) for \( \varepsilon \rightarrow 0 \)? The authors answer this problem in a profound study on weak convergence of \( \xi_{\varepsilon}(\nu_{\varepsilon}(t)) , \varepsilon\rightarrow 0\), in both topologies, the weak and the Skorokhod’s. Different models are treated separately, e.g., extremal processes with random sample size, randomly stopped extremal processes, extremal processes with asymptotically independent random sample size, extremal processes with conditionally independent random sample size, renewal-type and max-renewal-type extremal processes. These models are of interest in financial mathematics, environmetrics, telecommunication traffics and other typical applications of extreme value theory.
Reviewer: E.Pancheva (Sofia)

MSC:

60F05 Central limit and other weak theorems
60G70 Extreme value theory; extremal stochastic processes
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