The Mittag-Leffler distributions are two families of probability distributions on the half-line
[
0
,
∞
)
. They are parametrized by a real
α
∈
(
0
,
1
]
or
α
∈
[
0
,
1
]
. Both are defined with the Mittag-Leffler function, named after Gösta Mittag-Leffler.
For any complex
α
whose real part is positive, the series
E
α
(
z
)
:=
∑
n
=
0
∞
z
n
Γ
(
1
+
α
n
)
defines an entire function. For
α
=
0
, the series converges only on a disc of radius one, but it can be analytically extended to
C
−
{
1
}
.
The first family of Mittag-Leffler distributions is defined by a relation between the Mittag-Leffler function and their cumulative distribution functions.
For all
α
∈
(
0
,
1
]
, the function
E
α
is increasing on the real line, converges to
0
in
−
∞
, and
E
α
(
0
)
=
1
. Hence, the function
x
↦
1
−
E
α
(
−
x
α
)
is the cumulative distribution function of a probability measure on the non-negative real numbers. The distribution thus defined, and any of its multiples, is called a Mittag-Leffler distribution of order
α
.
All these probability distributions are absolutely continuous. Since
E
1
is the exponential function, the Mittag-Leffler distribution of order
1
is an exponential distribution. However, for
α
∈
(
0
,
1
)
, the Mittag-Leffler distributions are heavy-tailed. Their Laplace transform is given by:
E
(
e
−
λ
X
α
)
=
1
1
+
λ
α
,
which implies that, for
α
∈
(
0
,
1
)
, the expectation is infinite. In addition, these distributions are geometric stable distributions.
The second family of Mittag-Leffler distributions is defined by a relation between the Mittag-Leffler function and their moment-generating functions.
For all
α
∈
[
0
,
1
]
, a random variable
X
α
is said to follow a Mittag-Leffler distribution of order
α
if, for some constant
C
>
0
,
E
(
e
z
X
α
)
=
E
α
(
C
z
)
,
where the convergence stands for all
z
in the complex plane if
α
∈
(
0
,
1
]
, and all
z
in a disc of radius
1
/
C
if
α
=
0
.
A Mittag-Leffler distribution of order
0
is an exponential distribution. A Mittag-Leffler distribution of order
1
/
2
is the distribution of the absolute value of a normal distribution random variable. A Mittag-Leffler distribution of order
1
is a degenerate distribution. In opposition to the first family of Mittag-Leffler distribution, these distributions are not heavy-tailed.
These distributions are commonly found in relation with the local time of Markov processes. Parameter estimation procedures can be found here.
