According to the generalized Pólya theorem, the Gaussian distribution on the real line is characterized by the property of equidistribution of a monomial and a linear form of independent identically distributed random variables. We give a complete description of \({{\varvec{a}}}\)-adic solenoids for which an analog of this theorem is true. The proof of the main theorem is reduced to solving some functional equation in the class of continuous positive definite functions on the character group of an \({\varvec{a}}\)-adic solenoid.

中文翻译：

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关于 a-adic 电磁阀的广义 Pólya 定理

根据广义波利亚定理，实线上的高斯分布具有单项式的等分布性和独立同分布随机变量的线性形式。我们给出了\({{\varvec{a}}}\) -adic 螺线管的完整描述，对于该定理的模拟是正确的。主定理的证明简化为在\({\varvec{a}}\) -adic螺线管的字符组上求解连续正定函数类中的一些函数方程。