A complex-valued random variable Z is rotationally invariant if the moments of Z are the same as the moments of W=e^{i*theta}Z. In the first part of the article, we characterize such random variables, in terms of "vanishing unbalanced moments," moment and cumulant generating functions, and polar decomposition. In the second part, we consider random variables whose moments are not necessarily finite, but which have a density. In this setting, we prove two characterizations that are equivalent to rotational invariance, one involving polar decomposition, and the other involving entropy. If a random variable has both a density and moments which determine it, all of these characterizations are equivalent.

Author Bio

Michael Maiello is studying math at Texas A&M, College Station, and will be completing his Ph.D. at The University of Florida. His mathematical
interests include probability theory and geometric group
theory. In his spare time, he enjoys playing billiards, playing guitar, and rating

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