A limit theorem for selectors

Abstract

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 $X$.