Background on Real and Complex Elliptically Symmetric Distributions

This chapter presents a short overview of real elliptically symmetric (RES) distributionsReal elliptically symmetric (RES) distribution, complemented by circular complex elliptically symmetric (C-CES)Circular complex elliptically symmetric (C-CES) distribution and noncircular CES (NC-CES) distributionsNoncircular complex elliptically symmetric (NC-CES) distribution as complex representations of RES distributionsReal elliptically symmetric (RES) distribution. These distributions are both an extension of the multivariate Gaussian distributionGaussian distribution and a multivariate extension of univariate symmetric distributions. They are equivalently defined through their characteristic functionsCharacteristic function and their stochastic representationsStochastic representation, which naturally follow from the spherically symmetric distributions after affine transformationsAffine transformation. Particular attention is paid to the absolutely continuous case and to the subclass of compound Gaussian distributionsCompound Gaussian (CG) distribution. Results related to momentsMoment, affine transformationsAffine transformation, marginal and conditional distributionsConditional distribution, and summation stabilitySummation stability are also presented. Some well-known instances of RES distributionsReal elliptically symmetric (RES) distribution are provided with their main properties. Finally, the estimation of the symmetry centerSymmetry center and scatter matrixScatter matrix is briefly discussed through the sample meanSample mean (SM), sample covariance matrixSample covariance matrix (SCM) estimate, maximum estimate (ML), M-estimators, and Tyler’s M-estimatorsTyler’s M-estimator. Particular attention will be paid to the asymptotic Gaussianity of the M-estimators of the scatter matrixScatter matrix. To conclude, some hints about the Slepian–Bangs formulaSlepian–Bangs’s formula are provided.