We consider separately a spin and an oscillator that are coupled to their environment. After a finite interval of random length, the state of the environment changes, and each change causes a random change in the resonance frequency of the spin or vibrational frequency of the oscillator. Mathematically, the evolution of these frequencies is described by a continuous-time random walk. Physically, the stochastic dynamics can be understood as non-Gaussian because the frequency of the system and

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... te of the environment change on comparable time scales. These dynamics are also nonstationary, and so might apply to a nonequilibrium environment. The resonance and vibrational spectra of the spin and oscillator, as well as the ensemble-averaged displacement of the oscillator, are investigated in detail. We observe some distinct non-Gaussian features of the dynamics, such as the narrow, leptokurtic shape of the resonance spectrum of the spin and beating of the average oscillator displacement. The convergence to Gaussian dynamics as changes in the environment occur with increasing frequency is also considered. Among other results, we observe narrowing of the resonance and vibrational lines in the Gaussian limit due to a weakening of the system-environment interaction.