Limit laws for power normalized partial maxima of independent random variables having one of \(m\) specified distributions (Q2771452)

Let \(X_1,\dots ,X_n\) be independent r.v.'s with d.f.'s \(F_{X_n}\). Assume that among \(\{F_{X_n}\}\) only \(m\) are different, say \(F_1,\dots ,F_m\). Suppose that \(n_j(n)\) among \(X_1,\dots ,X_n\) have df \(F_j\). So \(\sum^m_{j=1} n_j(n) = n\). The so-called Gnedenko's problem (posed about five decades ago) consists of characterizing the class \({\mathcal G}_m\) of all limit distribution for properly normalized sums of r.v.'s in this setup. Obviously the class \({\mathcal G}_1\) coincides with the class of the stable distributions. The class \({\mathcal G}_2\) consists of compositions of pairs of stable df's [cf. \textit{V. M. Zolotarev} and \textit{V. S. Korolyuk}, Theory Probab. Appl. 6, 431-435 (1961); translation from Teor. Veroyatn. Primen. 6, 469-473 (1961; Zbl 0106.34101)]. \textit{A. A. Zinger} [ibid. 10, 607-626 (1965), resp. ibid. 10, 672-692 (1965; Zbl 0235.60021)] showed that for \(m \geq 3\) every df \(G \in {\mathcal G}_m\) can be represented as a convolution of \( s, s \leq m\), pseudo-stable laws. The latter are very natural generalizations of the stable laws. Their properties are studied e.g. by A. A. Zinger (loc. cit.), Yu. Khokhlov (1999). But a full description of the conditions under which one has a convergence to a given df \(G \in{\mathcal G}_m\) is still an open problem. There are only some particular results. The Gnedenko's problem in the (multivariate) extreme value setup seems to be firstly considered by Pancheva (1991). She describes the class max-\({\mathcal G}_2\) using monotone normalization. The present paper is devoted to the characterization of the class max-\({\mathcal G}_m\) for \(m \geq 2\) under the assumption that \(F_j\) belongs to the domain of attraction of a max-stable df \(H_j\), \(j=1,\dots, m\), and \(H_1,\dots ,H_m\) belong to the same type. The author uses power normalizations for compactifying the sequence \(M_n= \max (X_1,\dots ,X_n)\) and for defining the notion ``type'' of df's. He gives a criterion for the convergence of the power normalized sequence \(M_n\) if \(n_j(n)/n\rightarrow \alpha_j \in (0,1)\), \(1\leq j \leq m \).