- Source: Mittag-Leffler distribution
The Mittag-Leffler distributions are two families of probability distributions on the half-line
[
0
,
∞
)
{\displaystyle [0,\infty )}
. They are parametrized by a real
α
∈
(
0
,
1
]
{\displaystyle \alpha \in (0,1]}
or
α
∈
[
0
,
1
]
{\displaystyle \alpha \in [0,1]}
. Both are defined with the Mittag-Leffler function, named after Gösta Mittag-Leffler.
The Mittag-Leffler function
For any complex
α
{\displaystyle \alpha }
whose real part is positive, the series
E
α
(
z
)
:=
∑
n
=
0
∞
z
n
Γ
(
1
+
α
n
)
{\displaystyle E_{\alpha }(z):=\sum _{n=0}^{\infty }{\frac {z^{n}}{\Gamma (1+\alpha n)}}}
defines an entire function. For
α
=
0
{\displaystyle \alpha =0}
, the series converges only on a disc of radius one, but it can be analytically extended to
C
∖
{
1
}
{\displaystyle \mathbb {C} \setminus \{1\}}
.
First family of Mittag-Leffler distributions
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
]
{\displaystyle \alpha \in (0,1]}
, the function
E
α
{\displaystyle E_{\alpha }}
is increasing on the real line, converges to
0
{\displaystyle 0}
in
−
∞
{\displaystyle -\infty }
, and
E
α
(
0
)
=
1
{\displaystyle E_{\alpha }(0)=1}
. Hence, the function
x
↦
1
−
E
α
(
−
x
α
)
{\displaystyle x\mapsto 1-E_{\alpha }(-x^{\alpha })}
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
{\displaystyle \alpha }
.
All these probability distributions are absolutely continuous. Since
E
1
{\displaystyle E_{1}}
is the exponential function, the Mittag-Leffler distribution of order
1
{\displaystyle 1}
is an exponential distribution. However, for
α
∈
(
0
,
1
)
{\displaystyle \alpha \in (0,1)}
, the Mittag-Leffler distributions are heavy-tailed, with
E
α
(
−
x
α
)
∼
x
−
α
Γ
(
1
−
α
)
,
x
→
∞
.
{\displaystyle E_{\alpha }(-x^{\alpha })\sim {\frac {x^{-\alpha }}{\Gamma (1-\alpha )}},\quad x\to \infty .}
Their Laplace transform is given by:
E
(
e
−
λ
X
α
)
=
1
1
+
λ
α
,
{\displaystyle \mathbb {E} (e^{-\lambda X_{\alpha }})={\frac {1}{1+\lambda ^{\alpha }}},}
which implies that, for
α
∈
(
0
,
1
)
{\displaystyle \alpha \in (0,1)}
, the expectation is infinite. In addition, these distributions are geometric stable distributions. Parameter estimation procedures can be found here.
Second family of Mittag-Leffler 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
]
{\displaystyle \alpha \in [0,1]}
, a random variable
X
α
{\displaystyle X_{\alpha }}
is said to follow a Mittag-Leffler distribution of order
α
{\displaystyle \alpha }
if, for some constant
C
>
0
{\displaystyle C>0}
,
E
(
e
z
X
α
)
=
E
α
(
C
z
)
,
{\displaystyle \mathbb {E} (e^{zX_{\alpha }})=E_{\alpha }(Cz),}
where the convergence stands for all
z
{\displaystyle z}
in the complex plane if
α
∈
(
0
,
1
]
{\displaystyle \alpha \in (0,1]}
, and all
z
{\displaystyle z}
in a disc of radius
1
/
C
{\displaystyle 1/C}
if
α
=
0
{\displaystyle \alpha =0}
.
A Mittag-Leffler distribution of order
0
{\displaystyle 0}
is an exponential distribution. A Mittag-Leffler distribution of order
1
/
2
{\displaystyle 1/2}
is the distribution of the absolute value of a normal distribution random variable. A Mittag-Leffler distribution of order
1
{\displaystyle 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.