A statistical approach is introduced for describing influences of turbulence on high-frequency combustion instability in liquid-propellent rocket motors. The Rayleigh criterion is generalized to account for turbulence-related spatial variations of the combustion response. Special attention is given to the distinguished limit in which the acoustic oscillations are rapid compared with the turbulent fluctuations, which in turn are rapid compared with the linear growth rate of the instability. It is shown that nonlinear acoustics of the instabilities need to be considered to describe the influences of turbulence. When this nonlinearity involves supercritical bifurcation, the turbulence lends to decrease the most probable intensity of the instability. When it involves subcritical bifurcation, the turbulence can produce a bimodal probability density function for the intensity of the instability, with appreciable probabilities of high-amplitude and low-amplitude acoustic oscillations but small probabilities of oscillations of intermediate amplitudes. These phenomena can have a bearing on erratic pressure-amplitude bursts sometimes observed in liquid-propellant engines.