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} ,
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