Let
G
G
be the identity component of
S
O
(
n
,
1
)
\mathrm {SO}(n,1)
,
n
≥
2
n\ge 2
, acting linearly on a finite-dimensional real vector space
V
V
. Consider a vector
w
0
∈
V
w_0\in V
such that the stabilizer of
w
0
w_0
is a symmetric subgroup of
G
G
or the stabilizer of the line
R
w
0
\mathbb {R} w_0
is a parabolic subgroup of
G
G
. For any non-elementary discrete subgroup
Γ
\Gamma
of
G
G
with its orbit
w
0
Γ
w_0\Gamma
discrete, we compute an asymptotic formula (as
T
→
∞
T\to \infty
) for the number of points in
w
0
Γ
w_0\Gamma
of norm at most
T
T
, provided that the Bowen-Margulis-Sullivan measure on
T
1
(
Γ
∖
H
n
)
\mathrm {T}^1(\Gamma \backslash \mathbb {H}^n)
and the
Γ
\Gamma
-skinning size of
w
0
w_0
are finite.
The main ergodic ingredient in our approach is the description for the limiting distribution of the orthogonal translates of a totally geodesically immersed closed submanifold of
Γ
∖
H
n
\Gamma \backslash \mathbb {H}^n
. We also give a criterion on the finiteness of the
Γ
\Gamma
-skinning size of
w
0
w_0
for
Γ
\Gamma
geometrically finite.