Subexponentialiy of densities of infinitely divisible distributions

TL;DR

This paper establishes the equivalence of subexponential properties for densities and Lévy measures of infinitely divisible distributions under monotonicity conditions, broadening understanding and applications in statistical inference.

Contribution

It introduces novel conditions linking subexponentiality of densities and Lévy measures, and derives properties like closure under convolution, aiding statistical analysis.

Findings

Equivalence of subexponentiality between densities and Lévy measures.

Closure properties under convolution and asymptotic equivalence.

Applicability to statistical inference of continuous infinitely divisible distributions.

Abstract

We show the equivalence of three properties for an infinitely divisible distribution: the subexponentiality of the density, the subexponentiality of the density of its L\'evy measure and the tail equivalence between the density and its L\'evy measure density, under monotonic-type assumptions on the L\'evy measure density. The key assumption is that tail of the L\'evy measure density is asymptotic to a non-increasing function or is eventually non-increasing. Our conditions are novel and cover a rather wide class of infinitely divisible distributions. Several significant properties for analyzing the subexponentiality of densities have been derived such as closure properties of [ convolution, convolution roots and asymptotic equivalence ] and the factorization property. Moreover, we illustrate that the results are applicable for developing the statistical inference of subexponential infinitely divisible distributions which are absolutely continuous.