Near-source strong motions of 64 earthquakes (3 ≦ M ≦ 8.1; 10 ≦ R ≦ 54 km) recorded above the Mexican subduction zone are analyzed to study the scaling of peak horizontal acceleration, amax, and Fourier acceleration amplitude spectra, a(f), as a function of magnitude M. The amax data reduced to 16 km shows clear dependence on M for M ≦ 6. For M > 6, the existing data suggests that for such events amax does not depend on M. Observation points 16 km above the source are in the far field for earthquakes with M > 6 for the frequencies (>1 Hz) of interest here, so that for such events a(f) is easily explained in terms of Brune's source spectra modified by attenuation. The same spectra explain the amax data when Parseval's theorem is used to obtain arms and the expected amax is computed using random vibration theory (RVT). For larger earthquakes, we modify the point-source model to estimate Fourier amplitude spectra from finite sources, ignoring possible directivity effects. These spectra along with rupture duration Td is used to compute arms and to estimate amax by applying RVT. The character of the near-source recordings of 6 ≦ M ≦ 8.1 earthquakes in Mexico suggests that the assumption of stationarity over Td is reasonable. The results from the model show that beyond M ≅ 6, the dependence of amax on M decreases; for M > 7.5 amax becomes essentially independent of M. The amax and a(f) observed for M ≦ 6.5 may be interpreted in terms of this finite-source model with stress drops Δσ of 40 to 100 bars and an appropriate site attenuation parameter. From a possible M = 7.5 to 8 earthquake in the Guerrero gap the expected amax from the finite-source model in Acapulco, corresponding to Δσ = 100 bars and κ = 0.023s, is roughly g.