The authors analyze the effects of imperfect segment alignment on the aperture efficiency of a large millimeter telescope. A 50 meter diameter instrument of this type specified to operate to wavelengths as short as 1 mm is being designed with an actively controlled main surface. By simulating the performance of the control system, they generate samples of tilt and piston errors for the segments from which the antenna radiation patterns and aperture efficiencies are calculated. They make a comparison of these results with models of antenna tolerance theory developed by Ruze (1966), which relate the aperture efficiency to the RMS phase error. They find that Ruze's formulas have a different range of validity when the aperture RMS phase error, rather than the RMS surface error, is used as a parameter. When appreciable tilt errors are present in large segmented antennas, the aperture RMS phase error tends to a constant value, independent of the aperture illumination and of the shape of the segments. They conclude that the antenna RMS surface error is a better tracer of the aperture efficiency than is the aperture RMS phase error when Ruze's formula is used. They find that this well-known expression stands as a lower limit to the performance of large segmented reflector antennas.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>