In nonuniform sampling (NUS), signal amplitudes and time stamps are delivered in pairs. Several methods to compute an approximate Fourier transform (AFT) have appeared in literature, and their posterior properties in terms of alias suppression and leakage have been addressed. In this paper, the sampling times are assumed to be generated by a stochastic process. The main result gives the prior distribution of several AFTs expressed in terms of the true Fourier transform and variants of the characteristic function of the sampling time distribution. The result extends leakage and alias suppression with bias and variance terms due to NUS. Specific sampling processes as described in literature are analyzed in detail. The results are illustrated on simulated signals, with particular focus to the implications for spectral estimation.