Symmetrization of Bernoulli

Details Symmetrization of Bernoulli Symmetrization of Bernoulli

2006-01-26

arxiv.org/abs/math/0601652v1

Mathematics

Probability

3 pages; a completely probabilistic proof of a theorem due to Kagan, Mallows, Shepp, Vanderbei & Vardi

Scientific paper

Let X be a random variable. We shall call an independent random variable Y to be a symmetrizer for X, if X+Y is symmetric around zero. A random variable is said to be symmetry resistant if the variance of any symmetrizer Y, is never smaller than the variance of X itself. We prove that a Bernoulli(p) random variable is symmetry resistant if and only if p is not 1/2. This is an old problem proved in 1999 by Kagan, Mallows, Shepp, Vanderbei & Vardi using linear programming principles. We reprove it here using completely probabilistic tools using Skorokhod embedding and Ito's rule.

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