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New coins from old, smoothly. (English) Zbl 1238.41007
Given a coin with unknown probability of heads \(p\), as well as a fair coin, the authors would like to simulate a coin with probability of heads \(f(p)\), where \(f:[0,1]\to (0,1)\) is a known function. First, the authors define the simulation rate for a simulation algorithm. Next, they recall some basic results regarding Bernstein polynomials, Bernstein basis, Bernstein coefficients, Bernstein-positive consistent approximation from below, Bernstein-positive consistent approximation from above. The relationship between Bernstein-positive approximation and smoothness is then established. Next, Lorentz operators and simultaneous approximation are examined. An iterative construction of Bernstein-positive consistent approximations schemes is very clean presented. Finally, the authors prove that G. G. Lorentz’s Claim 10 [Math. Ann. 151, 239–251 (1963; Zbl 0116.04602)] is invalid.

41A10 Approximation by polynomials
41A25 Rate of convergence, degree of approximation
Full Text: DOI arXiv
[1] DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303. Springer, Berlin (1993)
[2] Keane, M.S., O’Brien, G.L.: A Bernoulli factory. ACM Trans. Model. Comput. Simul. 4(2), 213–219 (1994) · Zbl 0844.60008
[3] Lorentz, G.G.: The degree of approximation by polynomials with positive coefficients. Math. Ann. 151, 239–251 (1963) · Zbl 0116.04602
[4] Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea Publishing, New York (1986)
[5] Mossel, E., Peres, Y.: New coins from old: computing with unknown bias. Combinatorica 25(6), 707–724 (2005). With an appendix by C. Hillar · Zbl 1099.68052
[6] Nacu, Ş., Peres, Y.: Fast simulation of new coins from old. Ann. Appl. Probab. 15(1A), 93–115 (2005) · Zbl 1072.65007
[7] Peres, Y.: Iterating von Neumann’s procedure for extracting random bits. Ann. Stat. 20(1), 590–597 (1992) · Zbl 0754.60040
[8] 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
[9] 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 · Zbl 0188.00104
[10] Williams, D.: Probability with Martingales. Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge (1991)
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