Nonlinearities in power amplifiers adversely affect multi-carrier modulation techniques. Accurate prediction of nonlinear distortion is essential for making design trade-offs between output power and network throughput. We use the series form of the characteristic function (ch.f.) method to predict distortion spectra for sparse multi-carrier transmissions. This method results in efficient calculations of individual signal and distortion components. The method is validated both theoretically and practically. Theoretical validation is performed by modeling the signal as a bandpass Gaussian process that is hard limited, and it is shown that the series ch.f. method produces results that are identical with the classical Price's theorem. Practical validation is shown by considering an orthogonal frequency division multiplexing (OFDM) signal with a fragmented spectrum which is then applied to an amplifier driven into compression for which application of Price's theorem is difficult, and the predicted output spectrum corroborates laboratory measurements. Part of the computational efficiency is realized in that the nonlinearity can be expressed as the fast Fourier transform (FFT) of samples of its forward scattering parameter (i.e., S21) or transconductance function (including AM-PM effects), and distortion contributions of the signal can be expressed as numerical autoconvolutions of the clean spectrum. Signal-to-distortion ratio (SDR) can be easily computed and parameterized across variables of interest, such as overdrive level.