Abstract
The transformation of the plane which winds it up around the origin k times is called k-winding. We study invariance properties of probability measures under k-windings, in particular, relations with rotation invariance in the first part of the paper. Then winding versions of the Bernstein theorem on characterization of the product of normal distributions are obtained. Finally, it is shown that the second component of a 2-winding of iid variables does not identify distributions even of squares of the original variables. This fact is in a sharp contrast to the property of the first component, distribution of which does determine uniquely the distribution of iid variables.
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Acknowledgments
We are deeply indebted to Gérard Letac whose comments led to improvements in Sect. 2 and to V. Seshadri for discussions regarding Proposition 4.1. Thanks are also due to referees for valuable remarks. In particular, comments on proofs of Theorem 3.1 and Proposition 4.1 are much appreciated.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Misiewicz, J., Wesołowski, J. Winding planar probabilities. Metrika 75, 507–519 (2012). https://doi.org/10.1007/s00184-010-0339-z
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DOI: https://doi.org/10.1007/s00184-010-0339-z