On the Asymptotics of the Density of an Infinitely Divisible Distribution at Infinity

In this paper the asymptotic properties at infinity of the density of an infinitely divisible distribution are studied in the case where an absolutely continuous component of the Levy measure of this distribution varies dominantly at infinity. The presentation is given in terms of the so-called weak equivalence of functions which, in the case of weakly oscillating, and, in particular, the case of the density of an infinite divisible distribution regularly varying at infinity, coincides with ordinary equivalence.