We investigate the characteristics and limitations of the parameter τp(t), which represents the predominant period of the P wave. We analyze the effects of the length of the time window (TW), low-pass filter, and anelastic attenuation as well as the complexity of the source process to elucidate whether values of , or the time-dependent maximum value of τp(t), have direct relationships to the physical quantities of earthquakes. We find that values of have upper and lower limits. For larger earthquakes with source durations longer than the TW, the values of have an upper limit that depends on the TW. On the other hand, the values for smaller earthquakes have a lower limit that is proportional to the sampling interval. For intermediate earthquakes, the values of are close to their typical source durations and can have a large variety due to the complexity of the source process. These two limits and the slope for intermediate earthquakes yield an apparent final size dependence of in a wide magnitude range. As a result, the dependence of on the final size of earthquakes does not suggest that the final size of an earthquake is controlled by processes in the initial part of rupture. It is impossible to conclude whether the earthquake is deterministic or not from the dependence of on the final size of earthquakes. This is because does not always have a direct relation to the physical quantities of earthquakes.