# Mittag-Leffler function

To calculate the moments of the random variable *K _{n}/n^{a}* we need the generalized Mittag-Leffler distribution, g

_{a0}(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 *g _{a}* (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 *K _{n}/n^{a }*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*

and

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

where

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.