We consider the conditions for a finite set with a given system of operations (a finite algebra) to be subject to a probability limit theorem, i.e., arbitrary computations with mutually independent random variables have value distributions that tend to a certain limit (limit law) as the number of random variables used in the computation grows. Such behavior may be seen as a generalization of the central limit theorem that holds for sums of continuous random variables. We show that the existence of a limit probability law in a finite algebra has strong implications for its set of operations. In particular, with some geometric exceptions excluded, the existence of a limit law without zero components implies that all operations in the algebra are quasigroup operations and the limit law is uniform.

中文翻译：

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有限代数中概率极限定理的必要条件

我们考虑具有给定运算系统（有限代数）的有限集的条件服从概率极限定理，即具有相互独立的随机变量的任意计算具有趋向于某个极限（极限定律）的值分布随着计算中使用的随机变量数量的增加。这种行为可以看作是中心极限定理的推广，该定理适用于连续随机变量的总和。我们表明，有限代数中极限概率定律的存在对其操作集具有重要意义。特别是，排除了一些几何例外，没有零分量的极限定律的存在意味着代数中的所有运算都是拟群运算，并且极限定律是一致的。