Set parameters:

library(MittagLeffleR)
tail <- 0.7
scale <- 2
n <- 1000
cutoff <- 2 * scale

First type of Mittag-Leffler distribution

Generate 1000 i.i.d. random variables, and plot their empirical CDF (black) against the population CDF (red). The curved dashed line represents the stretched exponential function, due to the asymptotic result

\[f(x; \alpha, \tau) \sim \exp(-(x/\tau)^\alpha), \quad x \downarrow 0.\]

The straight dashed line in the second plot represents the function \(x^{-\alpha}\), showing asymptotic equivalence to a power-law for large values.

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A plot of the density:

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Second type of Mittag-Leffler distribution

The second type of Mittag-Leffler distribution is light-tailed, and in fact has finite moments of all orders: it drops off faster than the exponential distribution (dashed line).

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A plot of the density:

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