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 $X$.