## Abstract

We derive the representative Bernstein measure of the density of (X_{α})^{−α/(1−α)}, 0 < α < 1, where X_{α} is a positive stable random variable, as a Fox-H function. Up to a factor, this measure describes the law of some function aα of a uniform random variable U on (0,π). The distribution function of a_{1−α}(U) is then expressed through a H-function and is used to describe more explicitly the density of the analogue of Xα in the setting of free probability theory. Moreover, this density is shown to converge to a function with infinite mass when α → 0^{+}, in contrast to the classical setting where X_{α} is known to converge weakly to the inverse of an exponential random variable. Analytic evidences of the occurence of a_{α} in both the classical and the free settings conclude the exposition.

Original language | English (US) |
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Pages (from-to) | 137-149 |

Number of pages | 13 |

Journal | Electronic Communications in Probability |

Volume | 16 |

DOIs | |

State | Published - Jan 1 2011 |

## Keywords

- Fox H-function
- Free probability
- Stable laws

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty