Second order asymptotic behaviour of subordinated sequences with longtailed subordinator

Suppose that {a(n)} is a discrete probability distribution on the set N0 = {0, 1, 2, . . .} and {p(n)} is some non-negative sequence defined on the same set. The equation b(n) = ∞ k=0 p(k)a ∗k(n) defines a new sequence {b(n)}. Here {a ∗k(n)} denotes the k-fold convolution of the distribution {a(n)}. In the paper the asymptotic behaviour of the sequence {b(n)} is investigated. It is known that for the large classes of the sequences {a(n)} and {p(n)}, b(n) ∼ σp([σn]), where 1/σ is the mean of the distribution {a(n)}. Themain object of the present work is to estimate the difference b(n) − σp([σn]) for some classes of the sequences {a(n)} and {p(n)}. © 2006 Elsevier Inc. All rights reserved