Second-order regular variation inherited from Laplace–Stieltjes transforms

Abstract

ABSTRACTLet be the survival function of a non negative random variable satisfying that is of second-order regular variation with the regular variation index − α < 0 and the second-order parameter ρ < 0. Let ϕ(s) denote the Laplace–Stieltjes transform of F. For α ∈ (m, m + 1) and α − ρ ∈ (k, k + 1) with non negative integers m, k, if m = k, it is proved that the second-order regular variation of implies that of ( − 1)k(φ(k)(0) − φ(k)(1/ · )), while for m < k, the second-order regular variation of implies that of ( − 1)k + 1φ(k + 1)(1/ · ); for both cases, the second-order parameter ρ can be inherited. Moreover, we prove that the converses also hold true under mild conditions. As an application, we show that the second-order regular variation of implies that of with regularity conditions; this result proves the converse of the result established in Mao and Hu (2013). Finally, we prove that the second-order parameter can also be easily inherited by the Laplace–Stieltjes transform of the cumulative distribution function of its reciprocals.