A limit theorem for continuous selectors

Any (measurable) function K from Rn to R defines an operator K acting on random variables X by K(X) = K(X 1,..., X n), where the X j are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x 1, x 2,..., x n) ∈ {x 1, x 2,..., x n}. For each such continuous selector H (except for projections onto a single coordinate) there is a unique point ωH in the interval (0, 1) so that, for any random variable X, the iterates H (N) acting on X converge in distribution as N → ∞ to the ωH-quantile of X.