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Polya's Characterization Theorem for Complex Random Variables

A complex random variable can be Gaussian in either the narrow or the wide sense. It is observed that Gaussian random variables in the wide sense do not have the 2-stability property (which is well known for the real case), while in the narrow definition they do possess it. Moreover, it is proved that this property characterizes the class of complex Gaussian random variables in the narrow sense; no other complex random variable enjoys it.