| Stable probability distributions (SPDs) are generalizations of the familiar Gaussian distribution, but have infinite variance. Thus, statistical techniques relying on finite variance are inapplicable to SPDs. In particular, the correlation structure of multivariate SPDs is much more complex than a Gaussian; it is described by a measure on the sphere, called a spectral measure, and is poorly understood.;In this work, the relationship between multivariate SPDs and their spectral measures is illuminated, and tools are developed for the statistical analysis of multivariate SPDs. Methods from nonabelian harmonic analysis are applied to express the spectral measure using spherical Fourier series; this leads to an efficient and practical method for estimating spectral measures from empirical data, even in very high dimensions. Formulae are computed which relate the estimation error of a spectral measure back to the estimation error of the original SPD. These results are then applied to the identification and analysis of stable stochastic processes. |
