In this article, firstly, we prove that functions of the form $\varphi \left(x\right)=e^{cx}{I}_{(-\infty ,0)}\left(x\right)+e^{bx}{I}_{[0,+\infty )}\left(x\right)$, $c,b$ constants, are the only solutions to the integral equation $\varphi \left(x\right)={\int }_0^1\varphi \left(ux\right)\varphi \left((1-u)x\right)du$. This indeed gives the result of Van Asshe (1987) who used the Schwartz distribution theory to prove that for i.i.d $X$ and $Y$, $UX+(1-U)Y\stackrel{d}{=}X$ if and only if $X$ has a Cauchy distribution. Secondly, by looking into certain recursive integral equations involving characteristic functions, we prove that if for an $n\ge 2$, the random weight mean $U_{\left(1\right)}{X}_1+(U_{\left(2\right)}-U_{\left(1\right)})X_2+\cdots +(1-U_{(n-1)})X_n$ has a Cauchy distribution, then $X_1$ has a Cauchy distribution; random variables $X_1,\dots ,X_n$ are i.i.d, the random weights are the cuts of $(0,1)$ by a uniform sample. The multivariate analogue of this result is also provided.

在本文中，首先，我们证明了形式的函数$\phi \left(X\right)=e^{CX}{我}_{(-\infty ,0)}\left(X\right)+e^{bX}{我}_{[0,+\infty )}\left(X\right)$,$C,b$常数，是积分方程的唯一解$\phi \left(X\right)={\int }_0^1\phi \left(你X\right)\phi \left((1-你)X\right)d你$. 这确实给出了 Van Asshe (1987) 的结果，他使用 Schwartz 分布理论证明对于 iid$X$和$是$,$üX+(1-ü)是\stackrel{d}{=}X$当且仅当$X$具有柯西分布。其次，通过研究某些涉及特征函数的递归积分方程，我们证明如果对于一个$n\ge 2$, 随机权重均值${ü}_{\left(1\right)}{X}_1+({ü}_{\left(2\right)}-{ü}_{\left(1\right)})X_2+\cdots +(1-{ü}_{(n-1)})X_n$有一个柯西分布，那么$X_1$具有柯西分布；随机变量$X_1,\dots ,X_n$是 iid，随机权重是$(0,1)$通过一个统一的样本。还提供了该结果的多元类似物。