THE RATE OF CONVERGENCE FOR SUBEXPONENTIAL DISTRIBUTIONS AND DENSITIES

Abstract

A distribution function F on the nonnegative real line is called subexponential if limx↑∞(1-F*n(x)/(1 - F(x)) = n for all n ≥ 2, where F*n denotes the n‐fold Stieltjes convolution of F with itself. In this paper, we consider the rate of convergence in the above definition and in its density analogue. Among others we discuss the asymptotic behavior of the remainder term Rn (x) defined by Rn(x) = 1 - F*n(x) - n(1 - F(x)) and of its density analogue rn (x) = -(Rn (x))'. Our results complement and complete those obtained by several authors. In an earlier paper, we obtained results of the form n(x) = O(1)f(x)R(x), where f is the density of F and R(x) = ∫0x(1-F(y))dy. In this paper, among others we obtain asymptotic expressions of the form Rn(x)=2nR2(x) + O(1)(-f'(x))R2(x) where f' is the derivative of f.