Lower limits and equivalences for convolution tails

Abstract

Suppose $F$ is a distribution on the half-line $[0,\infty )$. We study the limits of the ratios of tails $\overline{F\ast F}(x)/\bar{F}(x)$ as $x\to \infty$. We also discuss the classes of distributions $\mathcal{S}$, $\mathcal{S}(\gamma )$ and ${\mathcal{S}}^{\ast }$.

Published at http://dx.doi.org/10.1214/009117906000000647 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (15)

• On lower limits and equivalences for distribution tails of randomly stopped sums
• Asymptotic results for tail probabilities of sums of dependent heavy-tailed random variables
• Asymptotic behavior of the generalized St. Petersburg sum conditioned on its maximum
• On closedness under convolution roots related to an infinitely divisible distribution in the distribution class L(γ)
• Some positive conclusions related to the Embrechts-Goldie' conjecture
• Spatial random field models based on Lévy indicator convolutions
• Long velocity tails in plasmas and gravitational systems
• The local asymptotic estimation for the supremum of a random walk with generalized strong subexponential summands
• A convergent kinetic equation for gravitational and Coulomb systems
• Asymptotics of randomly stopped sums in the presence of heavy tails
• Lower limits for distributions of randomly stopped sums
• Asymptotics of convolution with the semi-regular-variation tail and its application to risk
• On asymptotic scales of independently stopped random sums
• On a transformation between distributions obeying the principle of a single big jump
• Convolution and convolution-root properties of long-tailed distributions