## Abstract

The function φ^{θ}_{α}(t) =1/1 + e^{-iθsgnt}|t|^{α}, α ε (0, 2), θ ε (-π, π], is a characteristic function of a probability distribution iff |θ| ≤ min(πα/2, π - πα/2). This distribution is absolutely continuous; for θ = 0 it is symmetric. The latter case was introduced by Linnik in 1953 [13] and several applications were found later. The case θ ≠ 0 was introduced by Klebanov, Maniya, and Melamed in 1984 [9], while some special cases were considered previously by Laha [12] and Pillai [18]. In 1994, Kotz, Ostrovskii and Hayfavi [10] carried out a detailed investigation of analytic and asymptotic properties of the density of the distribution for the symmetric case θ = 0. We generalize their results to the non-symmetric case θ ≠ 0. As in the symmetric case, the arithmetical nature of the parameter a plays an important role, but several new phenomena appear.

Original language | English (US) |
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Pages (from-to) | 523-544 |

Number of pages | 22 |

Journal | Journal of Fourier Analysis and Applications |

Volume | 5 |

Issue number | 6 |

DOIs | |

State | Published - 1999 |

Externally published | Yes |

## Keywords

- Cauchy type integral
- Characteristic function
- Completely monotonicity
- Liouville numbers
- Plemelj-Sokhotskii formula
- Unimodality

## ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Applied Mathematics