Laws of large numbers for infinite means with mixing sequences

Abstract

In this paper, we consider the laws of large numbers with infinite means based on $\varphi$-mixing sequences. An exact weak law and a strong law are obtained for $\varphi$-mixing asymmetrical Cauchy random variables. It is also presented that the weak law cannot extend to a strong law. In addition, some simulations are presented to illustrate our results of the laws of large numbers.