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 \(\varDelta _{n}(x):=\max\{\sqrt{x(1-x)/n},1/n\}\). The key to the proof is a new result in approximation theory: Let \(\mathcal{B}^{+}_{n}\) 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 \(\sum_{n=1}^{\infty}F_{n}\) with \(F_{n}\in \mathcal{B}^{+}_{n}\) 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 \(\varphi_{n}\in\mathcal{B}^{+}_{n}\) 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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References
DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303. Springer, Berlin (1993)
Keane, M.S., O’Brien, G.L.: A Bernoulli factory. ACM Trans. Model. Comput. Simul. 4(2), 213–219 (1994)
Lorentz, G.G.: The degree of approximation by polynomials with positive coefficients. Math. Ann. 151, 239–251 (1963)
Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea Publishing, New York (1986)
Mossel, E., Peres, Y.: New coins from old: computing with unknown bias. Combinatorica 25(6), 707–724 (2005). With an appendix by C. Hillar
Nacu, Ş., Peres, Y.: Fast simulation of new coins from old. Ann. Appl. Probab. 15(1A), 93–115 (2005)
Peres, Y.: Iterating von Neumann’s procedure for extracting random bits. Ann. Stat. 20(1), 590–597 (1992)
Timan, A.F.: Theory of Approximation of Functions of a Real Variable. Dover, New York (1994). Translated from the Russian by J. Berry. Translation edited and with a preface by J. Cossar. Reprint of the 1963 English translation
von Neumann, J.: Collected Works. Vol. V: Design of Computers, Theory of Automata and Numerical Analysis. Pergamon Press, The Macmillan, New York (1963). General editor: A.H. Taub
Williams, D.: Probability with Martingales. Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge (1991)
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Communicated by Carl de Boor.
Research of O. Holtz was supported in part by a Center of Pure and Applied Mathematics grant at UC Berkeley and by the Sofja Kovalevskaja Research Prize of the Humboldt Foundation, Germany.
Research of F. Nazarov was supported in part by NSF grants DMS-0501067 and DMS-0800243. Research of Y. Peres was supported in part by NSF grant DMS-0605166.
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Holtz, O., Nazarov, F. & Peres, Y. New Coins from Old, Smoothly. Constr Approx 33, 331–363 (2011). https://doi.org/10.1007/s00365-010-9108-5
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DOI: https://doi.org/10.1007/s00365-010-9108-5
Keywords
- Simulation
- Approximation order
- Positive approximation
- Bernstein operator
- Lorentz operators
- Polynomial reproduction
- Smoothness
- Hölder class