Analytic and Asymptotic Properties of Non-Symmetric Linnik's Probability Densities

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.