New Coins from Old, Smoothly |
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Authors: | Olga Holtz Fedor Nazarov Yuval Peres |
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Affiliation: | (1) Department of Economic Psychology, Educational Psychology and Evaluation, Faculty of Psychology, University of Vienna, Universitaetsstrasse 7, 1010 Vienna, Austria |
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Abstract: | Given a (known) function f:[0,1]→(0,1), we consider the problem of simulating a coin with probability of heads f(p) by tossing a coin with unknown heads probability p, as well as a fair coin, N times each, where N may be random. The work of Keane and O’Brien (ACM Trans. Model. Comput. Simul. 4(2):213–219, 1994) implies that such a simulation scheme with the probability ℙ p (N<∞) equal to 1 exists if and only if f is continuous. Nacu and Peres (Ann. Appl. Probab. 15(1A):93–115, 2005) proved that f is real analytic in an open set S⊂(0,1) if and only if such a simulation scheme exists with the probability ℙ p (N>n) decaying exponentially in n for every p∈S. We prove that for α>0 noninteger, f is in the space C α [0,1] if and only if a simulation scheme as above exists with ℙ p (N>n)≤C(Δ n (p)) α , where varDelta n(x):=max{?{x(1-x)/n},1/n}varDelta _{n}(x):=max{sqrt{x(1-x)/n},1/n}. The key to the proof is a new result in approximation theory: Let B+nmathcal{B}^{+}_{n} be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree n. We show that a function f:[0,1]→(0,1) is in C α [0,1] if and only if f has a series representation ?n=1¥Fnsum_{n=1}^{infty}F_{n} with Fn ? B+nF_{n}in mathcal{B}^{+}_{n} and ∑ k>n F k (x)≤C(Δ n (x)) α for all x∈[0,1] and n≥1. We also provide a counterexample to a theorem stated without proof by Lorentz (Math. Ann. 151:239–251, 1963), who claimed that if some jn ? B+nvarphi_{n}inmathcal{B}^{+}_{n} satisfy |f(x)−φ n (x)|≤C(Δ n (x)) α for all x∈[0,1] and n≥1, then f∈C α [0,1]. |
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