Abstract: | Any (measurable) function K from Rn to R defines an operator K acting on random variables X by K(X) = K(X1,..., Xn), where the Xj are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x1, x2,..., xn) ∈ {x1, x2,..., xn}. 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. |