This paper presents the minimum bound of the mean-squared phase-error of a bang-bang (BB) clock-and-data recovery (CDR) circuit under the condition of random phase tracking. An analogy between the Kalman filter and a linearized BB CDR is utilized for the derivation. The effects of demultiplexing, loop latency, and granular jitter are considered in the analysis to reflect reality. The validity of the theoretical analysis is supported by behavioral time domain simulation results.