On Powers of the Characteristic Function

Abstract

Let $CH({\mathbb {R}})$ denote the family of all characteristic functions of probability measures on the real line ${\mathbb {R}}$. Given an integer $n>1$ and $f\in CH({\mathbb {R}})$, set $C_n(f)=\{g\in CH({\mathbb {R}}): g^n\equiv f^n\}$. The purpose of this paper is to study the structure of $C_n(f)$. This interest is inspired by the following question posed by N. G. Ushakov: Do there exist two different $f, g\in CH({\mathbb {R}})$ such that $f^n\equiv g^n$ for some odd integer $n>1$? We show that the answer to this question is yes. Moreover, there exists $f\in CH({\mathbb {R}})$, such that $C_n(f)$ is non-trivial for all integer $n>1$. We provide some sufficient conditions guaranteeing the triviality of $C_n(f)$. As a consequence of this, we see that several frequently used characteristic functions f generate the trivial classes $C_n(f)$ for all integer $n>1$.