On a characterization of idempotent distributions on discrete fields and on the field of p-adic numbers

Abstract

We prove the following theorem. Let $X$ be a discrete field, $ξ$ and $η$ be independent identically distributed random variables with values in $X$ and distribution $μ$. The random variables $S=ξ+η$ and $D=(ξ-η)^2$ are independent if and only if $μ$ is an idempotent distribution. A similar result is also proved in the case when $ξ$ and $η$ are independent identically distributed random variables with values in the field of $p$-adic numbers $\mathbf{Q}_p$, where $p>2$, assuming that the distribution $μ$ has a continuous density.