A simple analytical method to compute the point spread function for ground-layer compensation at large telescopes is developed. It is shown that a particular form of spatial filtering of high-altitude turbulence achieves very good PSF uniformity and symmetry over a given field. Wave-front sensing with a single low-altitude Rayleigh LGS can reach performance close to optimum at telescopes of medium aperture. Using 4234 real turbulence profiles measured on 21 nights at Cerro Pachon realistic statistics of ground-layer compensation are computed for the first time. The median FWHM resolution of an AO system with a Rayleigh beacon at 10 km and actuator pitch 0.4 m at 4.2 m telescope is 0."53, 0."31, and 0."22 at 0.5, 0.7, and 1.0 μm wavelength respectively. The median increase of the bightness in the center of stellar image over uncompensated seeing is 1.2, 1.7, and 2.4 magnitudes at those wavelengths.© (2004) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.