Analytic and asymptotic properties of non-symmetric Linnik's probability densities

Abstract

The function

$$\varphi _\alpha ^\theta (t) = \frac{1}{{1 + e^{ - i\theta \operatorname{s} gnt} \left| t \right|^\alpha }},\alpha \in (0,2),\theta \in ( - \pi ,\pi ]$$

, is a characteristic function of a probability distribution iff$\left| \theta \right| \leqslant \min (\tfrac{{\pi \alpha }}{2},\pi - \tfrac{{\pi \alpha }}{2})$. 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 α plays an important role, but several new phenomena appear.