SU Chun, HU Zhishui, CHEN Yu, LIANG Hanying
Let F(x) be a distribution function supported on [0,∞) with an equilibrium distribution function Fe(x). In this paper we pay special attention to the hazard rate function re(x) of Fe(x), which is also called the equilibrium hazard rate (E.H.R.) of F(x). By the asymptotic behavior of re(x) we give a criterion to identify F(x) to be heavy-tailed or light-tailed. Moreover, we introduce two subclasses of heavy-tailed distributions, i.e., M and M*, where M contains almost all the most important heavy-tailed distributions in the literature. Some further discussions on the closure properties of M and M* under convolution are given, showing that both of them are ideal heavy-tailed subclasses. In the paper we also study the model of independent difference ξ = Z - ?, where Z and ? are two independent and non-negative random variables. We give intimate relationships of the tail distributions of ξ and Z, as well as relationships of tails of their corresponding equilibrium distributions. As applications, we apply the properties of class M to risk theory. In the final, some miscellaneous problems and examples are laid, showing the complexity of characterizations on heavy-tailed distributions by means of re(x).
),define Hilbert spaces H1k(Ω, φ) called Sobolev-Hardy spaces with weight φ. As a corollary of this identity, Hardy s inequalities with weight φ in C1(
) follow. At last, by Hardy s inequalities with weight φ = 1, discuss the eigenvalue problem of the Laplace-Hardy operator with critical parameter (N - 2)2/4 in H11(Ω).