Analytic and Asymptotic Properties of Linnik′s Probability Densities .II.

Abstract

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