On the distribution of Randomly Signed Sums and Tomaszewski’s Conjecture

Abstract: A Rademacher sum $X$ is a random variable characterized by real numbers $a_1, \ldots, a_n$, and is equal to

$$X = a_1 x_1 + \ldots + a_n x_n,$$ where $x_1, \ldots, x_n$ are independent signs (uniformly selected from $\{-1, 1\}$).

A conjecture by Bogusław Tomaszewski, 1986: all Rademacher sums $X$ satisfy $$\textup{Pr}[ |X| \leq \sqrt {\textup{Var}(X)} ] \geq 1/2$$

We prove the conjecture, and discuss other ways in which Rademacher sums behave like normally distributed variables.

Joint work with Nathan Keller.

mathematical physicsanalysis of PDEsclassical analysis and ODEscombinatoricscomplex variablesfunctional analysisinformation theorymetric geometryoptimization and controlprobability

Audience: researchers in the topic

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