# Asymptotic results for certain weak dependent random variables

## TL;DR

This paper establishes almost optimal laws of large numbers, central limit theorems, and empirical process convergence for a broad class of weakly dependent random variables with covariance control.

## Contribution

It introduces a new framework for weak dependence that includes various dependence structures and proves fundamental limit theorems within this framework.

## Key findings

- Proved a Strong Law of Large Numbers with near-optimal convergence rates.
- Established a Central Limit Theorem and its functional version for the class.
- Derived an inequality relating joint and product distributions similar to Newman's inequality.

## Abstract

We consider a special class of weak dependent random variables with control on covariances of Lipschitz transformations. This class includes, but is not limited to, positively, negatively associated variables and a few other classes of weakly dependent structures. We prove a Strong Law of Large Numbers with a characterization of convergence rates, which is almost optimal, in the sense that it is arbitrarily close to the optimal rate for independent variables. Moreover, we prove an inequality comparing the joint distributions with the product distributions of the margins, similar to the well-known Newman's inequality for characteristic functions of associated variables. As a consequence, we prove a Central Limit Theorem together with its functional counterpart, and also the convergence of the empirical process for this class of weak dependent variables.