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 \(n>1\).

中文翻译：

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论特征函数的幂

令\（CH（{\ mathbb {R}} \\）表示实线\（{\ mathbb {R}} \）上概率度量的所有特征函数的族。给定整数\（n> 1 \）和\（f \ in CH（{\ mathbb {R}}）\），设置\（C_n（f）= \ {g \ in CH（{\ mathbb {R} }）：g ^ n \ equiv f ^ n \} \）。本文的目的是研究\（C_n（f）\）的结构。这种兴趣受到NG Ushakov提出的以下问题的启发：在CH（{\ mathbb {R}}）\中是否存在两个不同的\（f，g \），使得\（f ^ n \ equiv g ^ n \ ）以获得一些奇数\（n> 1 \）？我们证明这个问题的答案是肯定的。而且，存在\（f \ in CH（{\ mathbb {R}}）\）中，这样\（C_n（f）\）对于所有整数\（n> 1 \）都是不平凡的。我们提供一些足够的条件来保证\（C_n（f）\）的琐碎性。结果，我们看到几个常用的特征函数f为所有整数\（n> 1 \）生成平凡的类\（C_n（f）\）。