Abstract
Any (measurable) function K from Rn to R defines an operator K acting on random variables X by K(X) = K(X 1,..., X n), where the X j are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x 1, x 2,..., x n) ∈ {x 1, x 2,..., x n}. 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.
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Durango, F., Fernández, J.L., Fernández, P. et al. A limit theorem for continuous selectors. Isr. J. Math. 214, 983–994 (2016). https://doi.org/10.1007/s11856-016-1369-7
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DOI: https://doi.org/10.1007/s11856-016-1369-7