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Max domains of attraction of univariate and multivariate \(p\)-max stable laws. (English) Zbl 0774.60029
Let \(F\) be a distribution function (d.f.) on \(R^ d\), \(d\geq 1\). Suppose that there exist norming constants \(a_ n(i)>0\), \(b_ n(i)\), \(1\leq i\leq d\), \(n\geq 1\), and a d.f. \(K\) on \(R^ d\) with nondegenerate univariate marginals such that \[ \lim_{n\to\infty} F^ n(a_ n(i)x_ i+b_ n(i), \;1\leq i\leq d)=K(x), \qquad x=(x_ 1,\dots,x_ d)\in C(K),\tag{1} \] where \(C(K)\) is the set of all continuity points of \(K\). We call \(K\) as max stable d.f. under linear normalization or simply \(l\)-max stable d.f. If (1) holds, we write \(F\in D_ l(K)\) to indicate that \(F\) belongs to the max domain of attraction of \(K\) under linear normalization. A d.f. \(F\) on \(R^ d\) is said to belong to the max domain of attraction of a d.f. \(H\) on \(R^ d\) with nondegenerate univariate marginals under power normalization if there exist norming constants \(\alpha_ n(i)<0\) and \(\beta_ n(i)\), \(1\leq i\leq d\), \(n\geq 1\), such that \[ \lim_{n\to\infty} F^ n(\alpha_ n(i) | x_ i|^{\beta_ n(i)} \text{sgn}(x_ i),\;1\leq i\leq d)=H(x), \qquad x\in C(H),\tag{2} \] where \(\text{sgn}(x_ i)=-1\) if \(x_ i<0\), \(=0\) if \(x_ i=0\), and \(=1\) if \(x_ i>0\). We denote this as \(F\in D_ p(H)\). We call \(H\) as max stable d.f. under power normalization or simply \(p\)- max stable d.f. if (2) holds. We say that two d.f.’s \(F\) and \(G\) are of the same \(p\)-type if there exist \(A>0\) and \(B>0\) such that \(F(x)=G(A| x|^ B\text{ sgn}(x))\) for \(x\in R\).
It has been shown by E. Pancheva [Stability problems for stochastic models, Proc. 8th Int. Semin., Uzhgorod/USSR 1984, Lect. Notes Math. 1155, 284-309 (1985; Zbl 0572.62023)] that a univariate \(p\)-max stable d.f. can be a \(p\)-type of one of six d.f.’s, which are presented in the paper under review, too. The authors present necessary and sufficient conditions for a univariate d.f. to belong to \(D_ p(.)\) for each of the six \(p\)-max stable laws. They compare the max domains of attraction of univariate \(l\)-max stable laws with those of \(p\)-max stable laws and show that every d.f. attracted to an \(l\)-max stable law is necessarily attracted to some \(p\)-max stable law and that \(p\)-max stable laws, in fact, attract more.
Reviewer: Z.Rychlik (Lublin)

MSC:
60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
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