In probability theory and directional statistics, a wrapped Lévy distribution is a wrapped probability distribution that results from the "wrapping" of the Lévy distribution around the unit circle.
The pdf of the wrapped Lévy distribution is
f
W
L
(
θ
;
μ
,
c
)
=
∑
n
=
−
∞
∞
c
2
π
e
−
c
/
2
(
θ
+
2
π
n
−
μ
)
(
θ
+
2
π
n
−
μ
)
3
/
2
where the value of the summand is taken to be zero when
θ
+
2
π
n
−
μ
≤
0
,
c
is the scale factor and
μ
is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields:
f
W
L
(
θ
;
μ
,
c
)
=
1
2
π
∑
n
=
−
∞
∞
e
−
i
n
(
θ
−
μ
)
−
c
|
n
|
(
1
−
i
sgn
n
)
=
1
2
π
(
1
+
2
∑
n
=
1
∞
e
−
c
n
cos
(
n
(
θ
−
μ
)
−
c
n
)
)
In terms of the circular variable
z
=
e
i
θ
the circular moments of the wrapped Lévy distribution are the characteristic function of the Lévy distribution evaluated at integer arguments:
⟨
z
n
⟩
=
∫
Γ
e
i
n
θ
f
W
L
(
θ
;
μ
,
c
)
d
θ
=
e
i
n
μ
−
c
|
n
|
(
1
−
i
sgn
(
n
)
)
.
where
Γ
is some interval of length
2
π
. The first moment is then the expectation value of z, also known as the mean resultant, or mean resultant vector:
⟨
z
⟩
=
e
i
μ
−
c
(
1
−
i
)
The mean angle is
θ
μ
=
A
r
g
⟨
z
⟩
=
μ
+
c
and the length of the mean resultant is
R
=
|
⟨
z
⟩
|
=
e
−
c
