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Complex elliptically symmetric (CES) distributions constitute a flexible and broad class of distributions for many engineering applications and include the widely used complex Gaussian, complex -, complex generalized Gaussian and symmetric -stable distributions for example. Their careful statistical characterization is needed. Moreover, the mechanisms for generating random variables from these distributions are not well defined in literature. In this paper we provide such treatment in order to provide a better insight on their statistical properties and simplifying proofs and derivations. For example, insightful expressions for complex kurtosis coefficients of CES distributions are derived providing an interpretation on what complex kurtosis really measures. Also derived are asymptotic distributions of circularity measures, the sample circularity quotient and coefficient, assuming i.i.d. samples from an unspecified CES distribution. Also new Wald's type detectors of circularity are proposed that are valid within CES distributions with finite fourth-order moments. These results are accompanied with examples that are of interest in developing signal processing algorithms for complex-valued signals.