Higher quality level is generally perceived by customers as improved performance by assigning a correspondingly higher satisfaction score. Usually, the quality level is measured by process capability indices. The index is used to determine whether a production process is capable of producing items within a specified tolerance. Some useful process capability indices $C_p$, $C_{pk}$ and $C_{pm}$ have been widely used in six sigma industries to assess process performance. Most evaluations on process capability indices focus on point estimates, which may result in unreliable assessments of process performance. It is necessary to investigate their asymptotic correlationship among process capability indices $hat{C}_p$, $hat{C}_{pk}$ and $hat{C}_{pm}$. In this paper, we study their asymptotic correlationship for some process capability indices $hat{C}_p$, $hat{C}_{pk}$ and $hat{C}_{pm}$ under the normal process.