Limitations on the additivity of moments in line-profile analysis

The leading terms in the centroid-range and the variance-range curves are additive in the contributions made by the various sources of line displacement and line broadening. The intercept of the variance-range curve contains non-additive contributions inherent in the slow approach to zero of the functions convoluted in the line profile. Further terms in the series for these curves contain non-additive contributions that depend both on the functions convoluted and the range used; these terms approach zero as an inverse power of the range as this is increased.