Convergence Rate of Sample Mean for $\varphi$-Mixing Random Variables with Heavy-Tailed Distributions

TL;DR

This paper investigates how quickly the sample mean converges for dependent random variables with heavy tails, revealing that the main parts converge faster than the tail parts.

Contribution

It introduces a novel analysis of the convergence rate for $\

Findings

Convergence rate of the sample mean is established for $\

The main parts of the sample mean converge faster than the tail parts.

Provides a method to analyze dependent variables with infinite variance.

Abstract

This article studies the convergence rate of the sample mean for $\varphi$-mixing dependent random variables with finite means and infinite variances. Dividing the sample mean into sum of the average of the main parts and the average of the tailed parts, we not only obtain the convergence rate of the sample mean but also prove that the convergence rate of the average of the main parts is faster than that of the average of the tailed parts.