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Random array phase and range limitations are rigorously analyzed in a probabilistic sense using the mean value theorem. The elements of this model are assumed to form a uniformly distributed element population and are bounded to within a spherical volume. By this it is determined limitations in element location are not to exceed a tenth of a wavelength using a uniform distribution. Phase limitations are found similarly not to exceed the inverse of the four times the maximum aperture size using a von Mises distribution (circular normal or Tikhonov distribution). Significance of these results apply new fundamental limits for element location and phase shifter accuracy enclosed to within a uniformly distributed spherical volume.
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Buchanan et al. (2015) studied this question.