A note on powers of the characteristic function

TL;DR

This paper investigates when different characteristic functions have equal powers, providing conditions, estimates on the number of such functions, and characterizing cases where these estimates are exact.

Contribution

It offers new insights into the structure of characteristic functions with equal powers, including bounds on their quantity and specific characterizations.

Findings

For all integers n > 1, there exist distinct characteristic functions with equal n-th powers.

The paper provides estimates for the number of characteristic functions sharing the same n-th power.

It characterizes functions for which these estimates are tight.

Abstract

Let $CH(R)$ denote the family of characteristic functions of probability measures (distributions) on the real line $R$. We study the following question: given an integer $n>1$, do there exist two different $f, g\in CH(R)$ such that $ f^n\equiv g^n$? For positive even $n$, well-known examples answer this question in the affirmative. It turns out that the same is true also for any odd $n>1$. For $f\in CH(R)$ and integer $n>1$, set $C_n(f)=\{g\in CH(R): g^n\equiv f^n\}$. In this paper, we give an estimate for cardinality (or cardinal number) of $C_n(f)$. In addition, we describe such $f$ for which our estimate is sharp.