The rate of convergence for subexponential distributions

Abstract

A distribution functionF on the nonnegative real line is called subexponential if $$\mathop {\lim }\limits_{x \to \infty } \left( {1 - F^{*n} (x)} \right)/\left( {1 - F(x)} \right) = n, for all n \geqslant 2,$$ whereF *n denotes then-fold Stieltjes convolution ofF with itself. In this paper, we consider the rate of convergence in the above definition and we discuss the asymptotic behavior ofR n (x) defined byR n (x)=1−F *n (x)−n(1−F(x)). Our results complement those previously obtained by several authors. In this paper, we define several new classes of functions related to regular variation andO-regular variation. As a typical result, in one of our theorems we show thatR n (x)=O(1)f(x)R(x), wheref(x) is the density ofF andR(x)=∫ 0 (1−F(y))dy. We also discuss some applications.