Abstract Large-amplitude thermoacoustic oscillations are unwanted in gas turbines due to the detrimental damage to combustors. Thus predicting the onset of such oscillations and a better understanding of the unstable behaviors are important. In this work, the stochastic properties of thermoacoustic oscillations in the subthreshold region of thermoacoustic systems are investigated theoretically and numerically. The energy conversion from the unsteady heat release to sound is mainly achieved via two ways: one is caused by inherent turbulent fluctuations, and the other is due to the flame response to the acoustic waves. The turbulence-induced non-coherent heat release fluctuation is characterized by colored noise, and the coupling between the unsteady heat release and acoustic pressure is characterized by a 3rd order polynomial. Both stochastic averaging and stochastic normal form are utilized to approximate the responses of the system as a Markov process. The result shows that the correlation time τc and intensity D of the colored noise have significant but opposite effects on the dynamics of thermoacoustic oscillations, including the most probable amplitude, autocorrelation of the system as well as the correlation time. In addition, the resonance-like behaviors in signal-to-noise ratio (SNR) are observed, which denotes the emergence of coherence resonance (CR) in this annular combustion system. The optimal noise intensity, at which SNR is maximized, becomes sensitive to the variation of τc, as τc is larger than certain value. Lastly, the extent of the system coherent motions relies significantly on the proximity to the supercritical Hopf bifurcation point. Being closer to the stability boundary is found to increase the strength of system association. Meanwhile, SNR of the colored noise-induced motion becomes more distinguished, and the optimal noise intensity is shifted to a smaller value. This variation can serve as a precursor to predict the onset of thermoacoustic instability, as the correlation time remains unchanged.