To study statistical properties of the near-source radiation field, 35 horizontal accelerograms were selected, recorded at distances R7 Mexican earthquakes. For each one, the maximum-amplitude segment, assumedly containing direct S waves, was selected by an automatic procedure. Then two parameters were determined for this segment: the peak to rms amplitude ratio, or peak factor PF, and the slope K of the dependence of PF on log duration. These two values were compared to the corresponding values expected for a simplest stochastic record model, of a segment of the stationary Gaussian process. For about 40% of the records, both parameters show clear (above 2&sgr;) deviations from this model. Another record model was also tested, of quasi-stationary Gaussian process (with time-varying rms amplitude). With this end, the original records were ''stationarized,'' that is, modified to attain nearly constant rms amplitude, and then analyzed in the same manner as the original ones. However, for a distinct fraction of records, deviations persisted. The type of the deviations is characteristic for a ''heavy-tailed'' amplitude distribution, with the enhanced probability of large peaks. The degree of expression of this phenomenon seems to decay with distance. This observation suggests that source radiation, as it is generated at the fault, often is inherently non-Gaussian (heavy-tailed), but because of the scattering and multipathing, it quickly loses this property. One can expect this picture not to be specific for Mexican earthquakes only. From the viewpoint of stochastic strong motion simulation, the results imply that the PF value calculated from the stationary Gaussian model can underestimate average peak acceleration, on the average, by some 25%. In an analysis aimed at the recurrence of rare peaks, the underestimation may be even larger: among 35 records studied, five have peak factors more than 50% above the values expected for the stationary Gaussian model. ¿ American Geophysical Union 1996