{"id":357869,"date":"2017-01-25T14:20:42","date_gmt":"2017-01-25T22:20:42","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=357869"},"modified":"2018-10-16T20:01:43","modified_gmt":"2018-10-17T03:01:43","slug":"new-coins-old-smoothly","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/new-coins-old-smoothly\/","title":{"rendered":"New Coins From Old, Smoothly"},"content":{"rendered":"

Given a (known) function f<\/span>:<\/span>[<\/span>0<\/span>,<\/span>1<\/span>]<\/span>\u2192<\/span>(<\/span>0<\/span>,<\/span>1<\/span>)<\/span><\/span><\/span><\/span>, we consider the problem of simulating a coin with probability of heads f<\/span>(<\/span>p<\/span>)<\/span><\/span><\/span><\/span> by tossing a coin with unknown heads probability p<\/span><\/span><\/span><\/span>, as well as a fair coin, N<\/span><\/span><\/span><\/span> times each, where N<\/span><\/span><\/span><\/span> may be random. The work of Keane and O’Brien (1994) implies that such a simulation scheme with the probability $\\P_p(N<\\infty)$<\/span><\/span><\/span> equal to 1 exists iff f<\/span><\/span><\/span><\/span> is continuous. Nacu and Peres (2005) proved that f<\/span><\/span><\/span><\/span> is real analytic in an open set S<\/span>\u2282<\/span>(<\/span>0<\/span>,<\/span>1<\/span>)<\/span><\/span><\/span><\/span> iff such a simulation scheme exists with the probability $\\P_p(N>n)$<\/span><\/span><\/span> decaying exponentially in n<\/span><\/span><\/span><\/span> for every p<\/span>\u2208<\/span>S<\/span><\/span><\/span><\/span>. We prove that for \u03b1<\/span>><\/span>0<\/span><\/span><\/span><\/span> non-integer, f<\/span><\/span><\/span><\/span> is in the space C<\/span>\u03b1<\/span><\/span>[<\/span>0<\/span>,<\/span>1<\/span>]<\/span><\/span><\/span><\/span> if and only if a simulation scheme as above exists with $\\P_p(N>n) \\le C (\\Delta_n(p))^\\alpha$<\/span><\/span><\/span>, where $\\Delta_n(x)\\eqbd \\max \\{\\sqrt{x(1-x)\/n},1\/n \\}$<\/span><\/span><\/span>. The key to the proof is a new result in approximation theory:
\nLet $\\B_n$<\/span><\/span><\/span> be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree n<\/span><\/span><\/span><\/span>. We show that a function f<\/span>:<\/span>[<\/span>0<\/span>,<\/span>1<\/span>]<\/span>\u2192<\/span>(<\/span>0<\/span>,<\/span>1<\/span>)<\/span><\/span><\/span><\/span> is in C<\/span>\u03b1<\/span><\/span>[<\/span>0<\/span>,<\/span>1<\/span>]<\/span><\/span><\/span><\/span> if and only if f<\/span><\/span><\/span><\/span> has a series representation \u2211<\/span>\u221e<\/span>n<\/span>=<\/span>1<\/span><\/span><\/span><\/span>F<\/span>n<\/span><\/span><\/span><\/span><\/span> with $F_n \\in \\B_n$<\/span><\/span><\/span> and \u2211<\/span>k<\/span>><\/span>n<\/span><\/span><\/span><\/span>F<\/span>k<\/span><\/span>(<\/span>x<\/span>)<\/span>\u2264<\/span>C<\/span>(<\/span>\u0394<\/span>n<\/span><\/span>(<\/span>x<\/span>)<\/span>)<\/span>\u03b1<\/span><\/span><\/span><\/span><\/span> for all x<\/span>\u2208<\/span>[<\/span>0<\/span>,<\/span>1<\/span>]<\/span><\/span><\/span><\/span> and n<\/span>\u2265<\/span>1<\/span><\/span><\/span><\/span>. We also provide a counterexample to a theorem stated without proof by Lorentz (1963), who claimed that if some $\\phi_n \\in \\B_n$<\/span><\/span><\/span> satisfy |<\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span>\u2212<\/span>\u03d5<\/span>n<\/span><\/span>(<\/span>x<\/span>)<\/span>|<\/span><\/span><\/span>\u2264<\/span>C<\/span>(<\/span>\u0394<\/span>n<\/span><\/span>(<\/span>x<\/span>)<\/span>)<\/span>\u03b1<\/span><\/span><\/span><\/span><\/span> for all x<\/span>\u2208<\/span>[<\/span>0<\/span>,<\/span>1<\/span>]<\/span><\/span><\/span><\/span> and n<\/span>\u2265<\/span>1<\/span><\/span><\/span><\/span>, then f<\/span>\u2208<\/span>C<\/span>\u03b1<\/span><\/span>[<\/span>0<\/span>,<\/span>1<\/span>]<\/span><\/span><\/span><\/span>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Given a (known) function f:[0,1]\u2192(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 […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Constructive 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