Timing jitter is a limiting factor of the quality of microwave signals sampled by ultrafast optical pulse trains. Despite its central importance, an account that is both physically grounded and applicable to a broad class of problems and amenable to yielding closed-form expressions including timing jitter is lacking. We consider an optical-pulse train produced by an active- or hybrid-mode-locked laser with timing jitter originating in both the locking oscillator and the mode-locked laser. Our focus is on the power spectral density of a stationary random signal generated by an Ornstein-Uhlenbeck process sampled by the mode-locked laser. Based on an analytically tractable theoretical treatment, we find conditions under which the power spectral density of the sampled signal maintains reasonable fidelity with the original within the signal bandwidth and elucidate the rôles of the departure from periodicity in the sampling and the effect of the timing jitter on the sampled signal on the degradation of the PSD due to timing jitter. The effective number of bits (ENOB) associated with intrinsic properties of the underlying mode-locked laser are predicted to be large,i.e., the ENOB in practical devices is likely to be dominated by extrinsic noise and quantization error.