Elementary symmetric polynomials in random variables

The subject of the paper is the probability-theoretic properties of elementary symmetric polynomials sigma (k) of arbitrary degree k in random variables X (i) (i=1,2,...,m) defined on special subsets of commutative rings R (m) with identity of finite characteristic m. It is shown that the probability distributions of the random elements sigma(k) (X-1,...,X (m) ) tend to a limit when m ->infinity if X-1,...,X (m) form a Markov chain of finite degree mu over a finite set of states V, V subset of R-m , with positive conditional probabilities. Moreover, if all the conditional probabilities exceed a prescribed positive number alpha, the limit distributions do not depend on the choice of the chain.