A limit theorem for selectors

Any (measurable) function \(K\) from \(\mathbb{R}^n\) to \(\mathbb{R}\) defines an operator \(\mathbf{K}\) acting on random variables \(X\) by \(\mathbf{K}(X)=K(X_1, \ldots, X_n)\), where the \(X_j\) are independent copies of \(X\). The main result of this paper concerns selectors \(H\), continuous functions defined in \(\mathbb{R}^n\) and such that \(H(x_1, x_2, \ldots, x_n) \in \{x_1,x_2, \ldots, x_n\}\). For each such selector \(H\) (except for projections onto a single coordinate) there is a unique point \(\omega_H\) in the interval \((0,1)\) so that for any random variable \(X\) the iterates \(\mathbf{H}^{(N)}\) acting on \(X\) converge in distribution as \(N \to \infty\) to the \(\omega_H\)-quantile of \(X\).