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

Abstract

In 1953, Linnik introduced the probability density p α( x) defined by means of its characteristic function φ α(t) = 1 1 + |t| α , 0 < α < 2. Recently, this density has received several applications. In this paper, expansions of p α( x) into convergent series in terms of log| x|, | x| kα , | x| k ( k = 0, 1, 2, …) are obtained and the asymptotic behaviour of p α( 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.