Abstract
This note is concerned with implications of spherical analogues of the central limit theorem of probability in Euclidean space. In particular, it is concerned with the presumption that the analogy holds in terms of interpreting a special spherical limiting distribution, the hyperspherical Brownian distribution, as the distribution of the resultant rotation composed by a sequence of successive random rotations under similarly mild assumptions as applied in the central limit theorem for Euclidean space. This interpretation has been stressed at several instances to indicate the superiority of the spherical Brownian distribution for applications in texture component fit methods. Here it is shown, however, that this presumption is false. Thus, an explicit correspondence of the Brownian form of texture components and processes causing preferred crystallographic orientations cannot be inferred from a central limit type argument.
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Schaeben, H., Nikolayev, D.I. The Central Limit Theorem in Texture Component Fit Methods. Acta Applicandae Mathematicae 53, 59–87 (1998). https://doi.org/10.1023/A:1005979525038
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DOI: https://doi.org/10.1023/A:1005979525038