The limit behavior of elementary symmetric polynomials of IID random variables when their order tends to infinity

Let xi(1), xi(2),... be a sequence of i.i.d. random variables, and consider the elementary symmetric polynomial S-(k)(n) of order k = k(n) of the first n elements xi(1),..., xi(n) of this sequence. We are interested in the limit behavior of S-(k)(n) with an appropriate transformation if k(n)/n --> alpha, 0 < alpha < 1. Since k(n) --> infinity as n --> infinity, the classical methods cannot be applied in this case and new kinds of results appear. We solve the problem under some conditions which are satisfied in the generic case. The proof is based on the saddlepoint method and a limit theorem for sums of independent random vectors which may have some special interest in itself.