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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