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In this paper, we propose a nonlinear stochastic model for time-varying resonances where the instantaneous phase (and frequency) of a sinusoidal oscillation is allowed to vary proportionally to an α-stable self-similar stochastic processes. The main motivation of our work stems from previous experimental and theoretical evidence that speech resonances in fricative sounds can be modeled phenomenologically as AM-FM signals with randomly varying instantaneous frequencies and that several signal classes related to turbulent phenomena are self-similar 1/f processes. Our general approach is to model the instantaneous phase of an AM-FM resonance as a self-similar α-stable process. As a special case, this random phase model includes the class of random fractal signals known as fractional Brownian motion. We theoretically explore this random modulation model and analytically derive its autocorrelation and power spectrum. We also propose an algorithm to fit this model to arbitrary resonances with random phase modulation. Further, we apply the above ideas to real speech data and demonstrate that this model is suitable for resonances of fricative sounds.