Curiosities of characteristic functions
The author proves the following two theorems. Theorem 1. For each symmetric closed subset of the real line \(\mathbb{R}\) that contains the origin there exist characteristic functions \(\varphi(t)\) and \(\psi(t)\) such that \(\varphi(t)= \psi(t)\) on this set and \(\varphi(t)\neq \psi(t)\) otherwise. Theorem 2. Let \(\varphi\) be the characteristic function of an absolutely continuous distribution with a continuous, strictly positive density. Then there exists, for each \(a>0\), a characteristic function \(\psi(t)\) such that \(\varphi(t)= \psi(t)\) if \(t= 0\) or \(|t|> a\) and \(\varphi(t)\neq \psi(t)\) otherwise. An interesting open problem is formulated.
scientific article; zbMATH DE number 1740122