Symmetrization of binary random variables (Q1962612)

We say that a random variable \(Y\) is a symmetrizer of a given random variable \(X\) if the distribution of \(X+Y\) is symmetric about 0. It is called an independent symmetrizer if, in addition, it is independent of \(X\). A symmetrizer is called minimum-variance if among all such symmetrizers it has minimum variance. In general, taking \(Y\) to be independent of \(X\) and having the same distribution as \(-X\) clearly produces a symmetric sum, but it may not be of minimum variance. We say that a random variable \(X\) is symmetry resistant if such a \(Y\) is indeed a minimum-variance independent symmetrizer. The authors discuss the above notions with respect to the case where \(X\) is a Bernoulli random variable, i.e. \(P(X=1)= p\) and \(P(X=0)= q\), where \(0< p< 1\) and \(q= 1-p\). The following results are proved. Theorem 1: \(X\) is symmetry resistant if and only if \(p\neq 1/2\). Theorem 2: Allowing dependence between \(X\) and its symmetrizer, the minimum-variance symmetrizer has variance \(pq- (p\wedge q)/2\). Theorem 3: There exists a random variable for which the minimum-variance independent symmetrizer is not unique.