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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