On weak convergence of quasi-infinitely divisible laws

TL;DR

This paper introduces criteria for weak convergence of quasi-infinitely divisible laws, extending classical results for infinitely divisible laws by analyzing their spectral functions and providing new insights into their convergence behavior.

Contribution

It establishes new conditions linking weak convergence of quasi-infinitely divisible laws with convergence of their spectral functions, expanding the understanding beyond existing results.

Findings

Criteria connecting weak convergence with spectral function convergence

Extension of classical infinitely divisible law results

Complementary results to Lindner, Pan, and Sato (2018)

Abstract

We study a new class of so-called quasi-infinitely divisible laws, which is a wide natural extension of the well known class of infinitely divisible laws through the L\'evy--Khinchine type representations. We are interested in criteria of weak convergence within this class. Under rather natural assumptions, we state assertions, which connect a weak convergence of quasi-infinitely divisible distribution functions with one special type of convergence of their L\'evy--Khinchine spectral functions. The latter convergence is not equivalent to the weak convergence. So we complement known results by Lindner, Pan, and Sato (2018) in this field.