Probability laws with 1-stable marginals are 1-stable. (English) Zbl 0745.60013

Samorodnitsky, Gennady; Taqqu, Murad S.

Ann. Probab. 19, No. 4, 1777-1780 (1991).

The paper shows that a random vector \(X=(X_ 1,\dots,X_ d)\) is stable with index \(\alpha=1\) of stability iff all linear combinations \(\sum_{i=1}^ d C_ iX_ i\) are stable with respect to the same index. It is shown that the result is true for \(1<\alpha\leq 2\) but it is wrong for \(0<\alpha<1\). This settles a problem of R. M. Dudley and M. Kanter [Proc. Am. Math. Soc. 45, 245-252 (1974; Zbl 0297.60007)].

Reviewer: A.Janssen (Düsseldorf)

MSC:

60E07

Infinitely divisible distributions; stable distributions

Keywords:

stable measure; stable marginals; weak convergence

Citations:

Zbl 0297.60007