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

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)].

### MSC:

 6e+08 Infinitely divisible distributions; stable distributions

### Keywords:

stable measure; stable marginals; weak convergence

Zbl 0297.60007
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