Second-Order Subexponential Behavior of Subordinated Sequences

Suppose that \(\left\{ {a\left( n \right),n{\text{ }} \in {\text{ }}N_0 } \right\}\), \(\left\{ {b\left( n \right),n{\text{ }} \in {\text{ }}N_0 } \right\}\), and \(\left\{ {p\left( n \right),n{\text{ }} \in {\text{ }}N_0 } \right\}\) are three discrete probability distributions related by the equation (E): \(b\left( n \right) = \sum\nolimits_{k = 0}^\infty {p\left( k \right)} a^{*k} \left( n \right)\), where \(\left\{ {a^{*k} \left( n \right),n \in N_0 } \right\}\) denotes the k-fold convolution of \(\left\{ {a\left( n \right),n \in {\text{ }}N_{\text{0}} } \right\}\) In this paper, we investigate the relation between the asymptotic behaviors of \(a\)and \(b\). It turns out that, for wide classes of sequences \(a\) and \(p\), relation (E) implies that \(b\left( n \right)/a\left( n \right) \to {\text{E}}N\), where \({\text{E}}N\) is the mean of \(p\). The main object of this paper is to discuss the rate of convergence in this result. In our main results, we obtain O-estimates and exact asymptotic estimates for the difference \(b\left( n \right) - {\text{E}}N{\text{ }}a\left( n \right)\).