On Necessary Conditions of Probability Limit Theorems in Finite Algebras

• Published in: Doklady. Mathematics Vol. 102; no. 1; pp. 301 - 303
• Main Author: Yashunsky, A. D.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.07.2020; Springer Nature B.V
• Subjects: Algebra; Continuity (mathematics); Independent variables; Mathematics; Mathematics and Statistics; Random variables; Theorems; random variable; quasigroup; finite algebra; uniform distribution; limit theorem
• ISSN: 1064-5624; 1531-8362
• DOI: 10.1134/S1064562420040213

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.