ANALYTIC AND ASYMPTOTIC PROPERTIES OF LINNIKS PROBABILITY DENSITIES .1.

In 1953, Linnik introduced the probability density p(alpha)(x) defined by means of its characteristic function phi(alpha)(t) = 1/1 + \t\(alp ha), 0 < alpha < 2. Recently, this density has received several applic ations. In this paper, expansions of p(alpha)(x) into convergent serie s in terms of log\x\, \x\(k alpha), \x\(k) (k = 0, 1, 2,...) are obtai ned and the asymptotic behaviour of p(alpha)(x) at 0 and infinity is i nvestigated. These expansions and the asymptotic behaviour at 0 are qu ite distinct in the cases (i) 1/alpha is an integer, (ii) 1/alpha is a non-integer rational number, and (iii) alpha is an irrational number. The first part of the paper deals with preliminaries and case (i) of integer-valued 1/alpha. (C) 1995 Academic Press, Inc.