New Coins from Old, Smoothly

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 a"(TM) (p) (N < a) 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 SaS,(0,1) if and only if such a simulation scheme exists with the probability a"(TM) (p) (N > n) decaying exponentially in n for every paS. We prove that for alpha > 0 noninteger, f is in the space C (alpha) [0,1] if and only if a simulation scheme as above exists with a"(TM) (p) (N > n)a parts per thousand currency signC(Delta (n) (p)) (alpha) , where and a (k > n) F (k) (x)a parts per thousand currency signC(Delta (n) (x)) (alpha) for all xa[0,1] and na parts per thousand yen1. We also provide a counterexample to a theorem stated without proof by Lorentz (Math. Ann. 151:239-251, 1963), who claimed that if some satisfy |f(x)-phi (n) (x)|a parts per thousand currency signC(Delta (n) (x)) (alpha) for all xa[0,1] and na parts per thousand yen1, then faC (alpha) [0,1].