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We consider one of the possible generalizations of sums of independent random variable to the case of operations on a finite set, namely quasigroup "sums" that use quasigroup operations on a given finite set instead of the addition operation. For quasigroup "sums" that contain n independent identically distributed random variables we prove that the rate of convergence of distributions to uniform distribution is exponential in n. [ABSTRACT FROM AUTHOR]

We consider the transformations of random variables over a finite associative ring by the addition and multiplication operations. For arbitrary finite rings, we construct families of distribution algebras, which are sets of distributions closed over sums and products of independent random variables. [ABSTRACT FROM AUTHOR]

The paper is concerned with sets of Bernoulli distributions which are closed under substitutions of independent random variables into Boolean functions from a given set (an algebra of Bernoulli distributions). A description of all finite algebras of Bernoulli distributions is given. [ABSTRACT FROM AUTHOR]

Transformations of independent random variables over a finite field by read-once formulas are considered. Subsets of probability distributions that are preserved by read-once transformations are constructed. Also we construct a family of distributions that may be arbitrarily closely approximated by a read-once combination of independent identically distributed random variables, whose distributions have no zero components. [ABSTRACT FROM AUTHOR]