$\mathbf{G}$-Central limit theorems and $\mathbf{G}$-invariance principles for associated random variables

TL;DR

This paper develops a general theoretical framework for limit theorems and invariance principles for associated random variables, extending classical results to more general infinitely divisible laws and dependent data.

Contribution

It introduces a broad framework for G-central limit theorems and invariance principles for associated data, including non-Gaussian and non-Poisson cases, unifying and extending prior results.

Findings

Confirmed Gaussian limit results for associated variables.

Extended classical results to Poisson and non-stationary associated data.

Established a general approach for infinitely divisible limit laws.

Abstract

The investigation asymptotic limits on associated data mainly focused on limit theorems of summands of associated data and on the related invariance principles. In a series of papers, we are going to set the general frame of the theory by considering an arbitrary infinitely decomposable (divisible) limit law for summands and study the associated functional laws converging to L\'evy processes. The asymptotic frame of Newman (1980) is still used as a main tool. Detailed results are given when $G$ is a Gaussian law (as confirmation of known results) and when $G$ is a Poisson law. In the later case, classical results for independent and identically distributed data are extended to stationary and non-stationary associated data.