DO NON-STRICTLY STABLE LAWS ON POSITIVELY GRADUATED
SIMPLY CONNECTED NILPOTENT LIE GROUPS LIE IN THEIR OWN
DOMAIN OF NORMAL ATTRACTION?

Abstract: In the classical case of the real line, it is clear from the very definition that
non-degenerate stable laws always belong to their own domain of normal attraction. The
question if the analogue of this is also true for positively graduated simply connected
nilpotent Lie groups (a natural framework for the generalization of the concept of stability to
the non-commutative case) turns out to be non-trivial. The reason is that, in this case,
non-strict stability is defined in terms of generating distributions of continuous one-parameter
convolution semigroups rather than just for the laws themselves. We show that the answer is
affirmative for non-degenerate (not necessarily strictly) -dilation-stable laws on simply
connected step 2-nilpotent Lie groups (so, e.g., all Heisenberg groups and all so-called groups
of type H; cf. Kaplan [6]) if . The proof generalizes to positively
graduated simply connected Lie groups which are nilpotent of higher step if .

2000 AMS Mathematics Subject Classification: Primary: 60B15; Secondary: 22E25,
60F05.

Keywords and phrases: Domain of normal attraction, stable semigroup, simply
connected nilpotent Lie group, Heisenberg group, group of type H.