Characterizations on Heavy-tailed Distributions by Means of Hazard Rate

Let F(x) be a distribution function supported on [0,∞), with an equilibrium distribution function Fe(x). In this paper we shall study the function $$r_e(x)( - {\rm ln}{\overline F}_e ( x ))^\prime = {\overline F}( x )/\int_x^\infty {\overline F}( u )du $$, which is called the equilibrium hazard rate of F. By the limiting behavior of re(x) we give a criterion to identify F to be heavy-tailed or light-tailed. Two broad classes of heavy-tailed distributions are also introduced and studied.