Let
μ
1
,
μ
2
,
…
{\mu _1},{\mu _2}, \ldots
be regular probability measures on a locally compact Abelian group G such that
μ
=
μ
1
∗
μ
2
∗
⋯
=
lim
μ
1
∗
⋯
∗
μ
n
\mu = {\mu _1} \ast {\mu _2} \ast \cdots = \lim {\mu _1} \ast \cdots \ast {\mu _n}
exists (and is a probability measure). For arbitrary G, we derive analogues of the Lévy theorem on the existence of an atom for
μ
\mu
and of the “pure theorems” of Jessen, Wintner and van Kampen (dealing with discrete
μ
1
,
μ
2
,
…
{\mu _1},{\mu _2}, \ldots
) in the case
G
=
R
d
G = {R^d}
. These results are applied to the asymptotic distribution
μ
\mu
of an additive function
f
:
Z
+
→
G
f:{Z_ + } \to G
after generalizing the Erdös-Wintner result
(
G
=
R
1
)
(G = {R^1})
which implies that
μ
\mu
is an infinite convolution of discrete probability measures.