On convolution closure properties of subexponentiality approaching from densities

TL;DR

This paper investigates how subexponentiality properties of densities change under convolution, providing new conditions for preservation and counterexamples to previous assumptions about non-closedness.

Contribution

It introduces conditions under which subexponentiality is preserved or lost through convolution, and corrects past results with counterexamples and revised proofs.

Findings

Subexponentiality is not closed under convolution.

Conditions are identified that preserve or change subexponentiality.

Counterexamples challenge previous assumptions about non-closedness.

Abstract

Non-closedness of subexponentiality by the convolution operation is well-known. We go a step further and show that subexponentiality and non-subexponentiality are generally changeable by the convolution. We also give several conditions, by which (non-) subexponentiality is kept. Most results are given with densities, which are easily converted to those for distributions. As a by-product, we give counterexamples to several past results, which were used to derive the non-closedness of the convolution, and modify the original proof.