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}$(X)=K(X_1, \ldots, X_n)$,wherethe$X_j$areindependentcopiesof$X$. The main result of this paper concerns selectors $H$, continuous functions defined in ${\mathbb{R}}^n$ and such that $H(x_1, x_2, \ldots, x_n) \in {x_1,x_2, \ldots, x_n}$\mathrm{.}Foreachsuchselector$H$ (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 $N\to \mathrm{\infty }$ to the ${\omega }_H$-quantile of $X$.