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

This article studies the convergence rate of the sample mean for $\phi$-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.