In
probability theory , the Landau distribution
[1] is a
probability distribution named after
Lev Landau .
Because of the distribution's "fat" tail, the
moments of the distribution, such as mean or variance, are undefined. The distribution is a particular case of
stable distribution .
Definition
The
probability density function , as written originally by Landau, is defined by the
complex
integral :
p
(
x
)
=
1
2
π
i
∫
a
−
i
∞
a
+
i
∞
e
s
log
(
s
)
+
x
s
d
s
,
{\displaystyle p(x)={\frac {1}{2\pi i}}\int _{a-i\infty }^{a+i\infty }e^{s\log(s)+xs}\,ds,}
where a is an arbitrary positive
real number , meaning that the integration path can be any parallel to the imaginary axis, intersecting the real positive semi-axis, and
log
{\displaystyle \log }
refers to the
natural logarithm .
In other words it is the
Laplace transform of the function
s
s
{\displaystyle s^{s}}
.
The following real integral is equivalent to the above:
p
(
x
)
=
1
π
∫
0
∞
e
−
t
log
(
t
)
−
x
t
sin
(
π
t
)
d
t
.
{\displaystyle p(x)={\frac {1}{\pi }}\int _{0}^{\infty }e^{-t\log(t)-xt}\sin(\pi t)\,dt.}
The full family of Landau distributions is obtained by extending the original distribution to a
location-scale family of
stable distributions with parameters
α
=
1
{\displaystyle \alpha =1}
and
β
=
1
{\displaystyle \beta =1}
,
[2] with
characteristic function :
[3]
φ
(
t
;
μ
,
c
)
=
exp
(
i
t
μ
−
2
i
c
t
π
log
|
t
|
−
c
|
t
|
)
{\displaystyle \varphi (t;\mu ,c)=\exp \left(it\mu -{\tfrac {2ict}{\pi }}\log |t|-c|t|\right)}
where
c
∈
(
0
,
∞
)
{\displaystyle c\in (0,\infty )}
and
μ
∈
(
−
∞
,
∞
)
{\displaystyle \mu \in (-\infty ,\infty )}
, which yields a density function:
p
(
x
;
μ
,
c
)
=
1
π
c
∫
0
∞
e
−
t
cos
(
t
(
x
−
μ
c
)
+
2
t
π
log
(
t
c
)
)
d
t
,
{\displaystyle p(x;\mu ,c)={\frac {1}{\pi c}}\int _{0}^{\infty }e^{-t}\cos \left(t\left({\frac {x-\mu }{c}}\right)+{\frac {2t}{\pi }}\log \left({\frac {t}{c}}\right)\right)\,dt,}
Taking
μ
=
0
{\displaystyle \mu =0}
and
c
=
π
2
{\displaystyle c={\frac {\pi }{2}}}
we get the original form of
p
(
x
)
{\displaystyle p(x)}
above.
Properties
The approximation function for
μ
=
0
,
c
=
1
{\displaystyle \mu =0,\,c=1}
Translation: If
X
∼
Landau
(
μ
,
c
)
{\displaystyle X\sim {\textrm {Landau}}(\mu ,c)\,}
then
X
+
m
∼
Landau
(
μ
+
m
,
c
)
{\displaystyle X+m\sim {\textrm {Landau}}(\mu +m,c)\,}
.
Scaling: If
X
∼
Landau
(
μ
,
c
)
{\displaystyle X\sim {\textrm {Landau}}(\mu ,c)\,}
then
a
X
∼
Landau
(
a
μ
−
2
a
c
log
(
a
)
π
,
a
c
)
{\displaystyle aX\sim {\textrm {Landau}}(a\mu -{\tfrac {2ac\log(a)}{\pi }},ac)\,}
.
Sum: If
X
∼
Landau
(
μ
1
,
c
1
)
{\displaystyle X\sim {\textrm {Landau}}(\mu _{1},c_{1})}
and
Y
∼
Landau
(
μ
2
,
c
2
)
{\displaystyle Y\sim {\textrm {Landau}}(\mu _{2},c_{2})\,}
then
X
+
Y
∼
Landau
(
μ
1
+
μ
2
,
c
1
+
c
2
)
{\displaystyle X+Y\sim {\textrm {Landau}}(\mu _{1}+\mu _{2},c_{1}+c_{2})}
.
These properties can all be derived from the characteristic function.
Together they imply that the Landau distributions are closed under
affine transformations .
Approximations
In the "standard" case
μ
=
0
{\displaystyle \mu =0}
and
c
=
π
/
2
{\displaystyle c=\pi /2}
, the pdf can be approximated
[4] using
Lindhard theory which says:
p
(
x
+
log
(
x
)
−
1
+
γ
)
≈
exp
(
−
1
/
x
)
x
(
1
+
x
)
,
{\displaystyle p(x+\log(x)-1+\gamma )\approx {\frac {\exp(-1/x)}{x(1+x)}},}
where
γ
{\displaystyle \gamma }
is
Euler's constant .
A similar approximation
[5] of
p
(
x
;
μ
,
c
)
{\displaystyle p(x;\mu ,c)}
for
μ
=
0
{\displaystyle \mu =0}
and
c
=
1
{\displaystyle c=1}
is:
p
(
x
)
≈
1
2
π
exp
(
−
x
+
e
−
x
2
)
.
{\displaystyle p(x)\approx {\frac {1}{\sqrt {2\pi }}}\exp \left(-{\frac {x+e^{-x}}{2}}\right).}
Related distributions
The Landau distribution is a
stable distribution with stability parameter
α
{\displaystyle \alpha }
and skewness parameter
β
{\displaystyle \beta }
both equal to 1.
References
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint)
Directional
Degenerate and
singular Families