On the distribution of randomly signed sums and Tomaszewski鈥檚 conjecture
On the distribution of randomly signed sums and Tomaszewski鈥檚 conjecture
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Ohad Klein, Bar Ilan University
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A Rademacher sum X is a random variable characterized by real numbers a_1,..., a_n, and is equal to X = a_1 x_1 + ... + a_n x_n, where x_1, ..., x_n are independent signs (uniformly selected from {-1, 1}). A conjecture by Bogus艂aw听Tomaszewski, 1986: All Rademacher sums X satisfy Pr[ |X| <= sqrt Var(X) ] >= 1/2. We prove the conjecture, and discuss other ways in which Rademacher sums behave like normally distributed variables. Joint work with Nathan Keller.