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.
