Abstract
The function
, 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.
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Erdogan, M.B. Analytic and asymptotic properties of non-symmetric Linnik's probability densities. The Journal of Fourier Analysis and Applications 5, 523–544 (1999). https://doi.org/10.1007/BF01257189
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DOI: https://doi.org/10.1007/BF01257189