We investigate numerically the statistical properties of a model of modulated diffusion for which we have already computed analytically the diffusion coefficient D. Our model is constructed by adding a deterministic or random noise to the frequency of an integrable isochronous system. We consider in particular the central limit theorem and the invariance principle and we show that they follow whenever D is positive and for any magnitude of the noise; we also investigate the asymptotic distribution in a case when D = 0.