Local subexponentiality and infinitely divisible distributions

TL;DR

This paper provides a comprehensive characterization of local and delta subexponentiality for positive-half and two-sided infinitely divisible distributions, extending existing theory and exploring their properties and applications.

Contribution

It introduces new conditions for delta- and local subexponentiality of infinitely divisible distributions, including two-sided cases, and explores their closedness properties and applications.

Findings

Characterization of delta- and local subexponentiality for positive-half distributions

Extension of subexponentiality characterization to two-sided distributions

Application to supremum of random work and stopped iid sums

Abstract

We completely characterize $\Delta$- and local subexponentialities of positive-half compound Poisson distributions and extend the characterization on two-sided distributions. Moreover, $\Delta$-subexponentiality of infinitely divisible distributions is characterized with new conditions, and local subexponentiality is newly characterized in the two-sided case. In the process closedness properties of these subexponentialities are derived, particularly for distributions on $\R$. Most results are obtained by exploiting monotonic-type assumptions. We apply our results to distributions of supremum of a random work and a randomly stopped iid sum.