Given a (known) function \\(f:[0,1]\\to (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 (1994) implies that such a simulation scheme with the probability \\(\\P_p(N<\\infty)\\) equal to 1 exists iff \\(f\\) is continuous. Nacu and Peres (2005) proved that \\(f\\) is real analytic in an open set \\(S\\subset (0,1)\\) iff such a simulation scheme exists with the probability \\(\\P_p(N>n)\\) decaying exponentially in \\(n\\) for every \\(p\\in S\\). We prove that for \\(\\alpha >0\\) non-integer, \\(f\\) is in the space \\(C_{\\alpha }[0,1]\\) if and only if a simulation scheme as above exists with \\(\\P_p(N>n) \\le C (\\Delta_n(p))^\\alpha\\), where \\(\\Delta_n(x)\\eqbd \\max \\{\\sqrt{x(1-x)\/n},1\/n \\}\\). The key to the proof is a new result in approximation theory: Let \\(\\B_n\\) be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree \\(n\\). We show that a function \\(f:[0,1]\\to (0,1)\\) is in \\(C_{\\alpha }[0,1]\\) if and only if \\(f\\) has a series representation \\({\\sum }_{\\infty }^{n=1}F_n\\) with \\(F_n \\in \\B_n\\) and \\({\\sum }_{k>n}F_k(x)\\leq C({\\Delta }_n(x))_{\\alpha }\\) for all \\(x\\in [0,1]\\) and \\(n\\geq 1\\). We also provide a counterexample to a theorem stated without proof by Lorentz (1963), who claimed that if some \\(\\phi_n \\in \\B_n\\) satisfy \\(|f(x)-{\\phi }_n(x)|\\leq C({\\Delta }_n(x))_{\\alpha }\\) for all \\(x\\in [0,1]\\) and \\(n\\geq 1\\), then \\(f\\in C_{\\alpha }[0,1]\\).<\/p>\n","protected":false},"excerpt":{"rendered":"
Given a (known) function , we consider the problem of simulating a coin with probability of heads by tossing a coin with unknown heads probability , as well as a fair coin, times each, where may be random. 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