We here present a comparative study of frequency stabilities of pump and probe lasers coupled at a frequency offset generated by coherent photon-atom interaction. Pump-probe spectroscopy of the Ù configuration in D₂ transition of cesium is carried out to obtain sub-natural (~ 2 MHz) electromagnetically induced transparency (EIT) and sub-Doppler (~ 10 MHz) Autler-Townes (AT) resonance. The pump laser is locked on the saturated absorption spectrum (SAS, ~ 13 MHz) and the probe laser is successively stabilized on EIT and AT signals. Frequency stabilities of pump and probe lasers are calculated in terms of Allan variance s (2,t) by using the frequency noise power spectrum. It is found that the frequency stability of the probe stabilized on EIT is superior (s ~ 2x10^-13)) to that of SAS locked pump laser (s ~ 10^-12)), whereas the performance of the AT stabilized laser is inferior (s ~ 6x10^-12)). This contrasting behavior is discussed in terms of the theme of conventional master-slave offset locking scheme and the mechanisms underlying the EIT and sub-Doppler AT resonances in a Doppler broadened atomic medium.