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Analytic Description of Limit Laws for Sequences of Random Symmetric Polynomials

The following problem is considered. There is a sequence of linearly normalized random symmetric polynomials of fixed degree k Z n =(Q nk (X 1 ,...,X n )−u n )/v n , formed using a sequence of independent random variables, X 1 ,X 2 ,..., with the same distribution function F. It is assumed that Z n converges in distributions as n→∞ to a nondegenerate r.v. Z. The question raised is what is the class F k of possible distributions of the r.v. Z