Analytic and Asymptotic Properties of Linnik′s Probability Densities ,I

In 1953, Linnik introduced the probability density pα(x) defined by means of its characteristic function φα(t) = 11 + |t|α, 0 < α < 2. Recently, this density has received several applications. In this paper, expansions of pα(x) into convergent series in terms of log|x|, |x|kα, |x|k (k = 0, 1, 2, …) are obtained and the asymptotic behaviour of pα(x) at 0 and ∞ is investigated. These expansions and the asymptotic behaviour at 0 are quite distinct in the cases (i) 1/α is an integer, (ii) 1/α is a non-integer rational number, and (iii) α is an irrational number. The first part of the paper deals with preliminaries and case (i) of integer-valued 1/α.