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Mittag-Leffler function

To calculate the moments of the random variable Kn/na we need the generalized Mittag-Leffler distribution, ga0(x).

To do this, we first need to define the Mittag-Leffler function,

and the generalized version is

The first two moments of the random variable L are and

From these we obtain the variance of L as where

The Mittag-Leffler function ga (x) has the property that its pth moment is given by

for p > —1. (See the Appendix.) Then the coefficient of variation of L is seen to become zero for a = 0.

Yamato and Sibuya (2000) and Pitman (2002) have shown that Kn/na converges to L in distribution, and almost surely, and

In discussing limiting expressions in the preceeding sections we encounter a random variable L with the density function called the Mittag-Leffler function. It is given by

The p-th moment of L is given by where P > —1.

Using this formula, we obtain the first two moments of the random variable L


From these we calculate the variance of L, and its coefficient of variation as


Note that the expression ^(a, в) defined above is zero at a = 0, otherwise it is positive for a > 0. See Appendix for some additional information on the Mittag-Leffler functions and distributions.

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