Let
X
1
,
X
2
,
…
{X_1},{X_2}, \ldots
be a sequence of independent indentically distributed stable random variables with parameters
α
(
0
>
α
>
2
)
\alpha \;(0 > \alpha > 2)
and
β
(
|
β
|
⩽
1
)
\beta (|\beta | \leqslant 1)
. Let
S
n
=
∑
i
=
1
n
X
i
{S_n} = \sum \nolimits _{i = 1}^n {{X_i}}
. Suppose that
(
S
1
,
n
)
({S_{1,n}})
and
(
S
2
,
n
)
({S_{2,n}})
are independent copies of the sequence
(
S
n
)
({S_n})
. In this paper we obtain the set of all limit points in the plane of the sequence
\[
{
|
n
−
1
/
α
(
S
1
,
n
−
a
n
)
|
1
/
(
log
log
n
)
,
|
n
−
1
/
α
(
S
2
,
n
−
a
n
)
|
1
/
(
log
log
n
)
}
\left \{ {|{n^{ - 1/\alpha }}({S_{1,n}} - {a_n}){|^{1/(\log \log n)}},|{n^{ - 1/\alpha }}({S_{2,n}} - {a_n}){|^{1/(\log \log n)}}} \right \}
\]
where
(
a
n
)
({a_n})
is zero if
α
≠
1
\alpha \ne 1
and is
(
2
β
n
log
n
)
/
π
(2\beta n\log n)/\pi
if
α
=
1
\alpha = 1
.