New Coins from Old, Smoothly

Abstract

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 $\sum_{n=1}^\infty F_n$ with $F_n \in \B_n$ and $\sum_{k>n} F_k(x) \le C(\Delta_n(x))^\alpha$ for all $ x \in [0,1]$ and $n \ge 1$. 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 $|f(x)-\phi_n(x)| \le C (\Delta_n(x))^\alpha$ for all $ x \in [0,1]$ and $n \ge 1$, then $f \in C^\alpha [0,1]$.