The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions

TL;DR

This paper proves that the limits of nested subclasses of five key classes of infinitely divisible distributions on R^d coincide with the closure of stable distributions, extending previous results to more general cases.

Contribution

It establishes that the limits of nested subclasses of five important classes of infinitely divisible distributions are identical with the closure of stable distributions, including more general cases.

Findings

Limits of nested subclasses match the closure of stable distributions.

Results apply to five major classes of infinitely divisible distributions.

Provides more general conditions for the equivalence.

Abstract

It is shown that the limits of the nested subclasses of five classes of infinitely divisible distributions on $R^d$, which are the Jurek class, the Goldie-Steutel-Bondesson class, the class of selfdecomposable distributions, the Thorin class and the class of generalized type $G$ distributions, are identical with the closure of the class of stable distributions. More general results are also given.