Boolean and Free Symmetrization of Bernoulli Laws

Abstract: We revisit the notion of \emph{symmetrizers} for Bernoulli random variables, originally studied in the classical convolution setting by Harremo\"es and Vignat \cite{HarremoesVignat2008} and later refined in Dossani \cite{Dossani2008}. In the classical case, an asymmetric Bernoulli distribution is \emph{symmetry resistant}: any independent symmetrizer must have variance at least $pq$, where $p$ is the Bernoulli parameter and $q=1-p$. We prove an analogous result for \emph{Boolean convolution}, showing that the same lower bound persists in this noncommutative framework. Our method is based on the Boolean $K$--transform, which linearizes Boolean additive convolution and admits a natural symmetry criterion. We also discuss the case of \emph{free convolution}, proving existence of a free symmetrizer with variance $pq$ and formulating the minimality question as an open problem.