Abnormally heavy precipitation events can lead to numerous hazards, including flooding, landslides, and avalanches. Their developments require a sufficient supply of moisture and some physical mechanism to produce condensation. Atmospheric rivers (ARs) defined as long and narrow corridors of strong horizontal moisture transport can provide such necessary conditions. The presence and strength of ARs are often described using the integrated water vapor (IWV) and the integrated vapor transport (IVT). However, the associated precipitation is not directly correlated with these two variables. It is the net convergence of moisture that determines the intensity of precipitation. The purpose of this study is to illustrate, in the context of AR analysis, how the converged vapor should be distributed between condensation and air moistening. A simple algorithm is proposed for estimating the heavy precipitation attributable to the IVT convergence. Bearing a strong resemblance to the Kuo-Anthes parameterization scheme for cumulus convection, the proposed algorithm calculates the large-scale primary condensation rate (PCR) as a proportion of the IVT convergence, with a reduction to account for the general moistening in the atmosphere. The amount of reduction is determined by the column relative humidity (CRH), which is defined as the ratio of IWV to its saturation counterpart. It is found that the PCR in an air column with CRH < 0.60 is negligibly small. Based on a one-year dataset from the Canadian global numerical weather prediction (NWP) model, the best cut-off value of CRH for the algorithm is 0.66. It is demonstrated that this diagnosable PCR compares well to the forecast precipitation rate given by the NWP model. Case studies are conducted to illustrate the usefulness of CRH and PCR as two complements to standard AR analysis and impact-based storm classification.