# An asymptotically optimal Bernoulli factory for certain functions that can be expressed as power series

Mendo Tomás, Luis (2019). An asymptotically optimal Bernoulli factory for certain functions that can be expressed as power series. "Stochastic Processes and their Applications", v. 129 (n. 11); pp. 4366-4384. ISSN 0304-4149. https://doi.org/10.1016/j.spa.2018.11.017.

## Description

Title: An asymptotically optimal Bernoulli factory for certain functions that can be expressed as power series Mendo Tomás, Luis Article Stochastic Processes and their Applications November 2019 0304-4149 129 Bernoulli factory; simulation; power series E.T.S.I. Telecomunicación (UPM) Señales, Sistemas y Radiocomunicaciones Tecnologías de la Información y las Comunicaciones GTIC Recognition

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## Abstract

Given a sequence of independent Bernoulli variables with unknown parameter $p$, and a function $f$ expressed as a power series with non-negative coefficients that sum to at most $1$, an algorithm is presented that produces a Bernoulli variable with parameter $f(p)$. In particular, the algorithm can simulate $f(p)=p^a$, $a\in(0,1)$. For functions with a derivative growing at least as $f(p)/p$ for $p\rightarrow 0$, the average number of inputs required by the algorithm is asymptotically optimal among all simulations that are fast in the sense of Nacu and Peres. A non-randomized version of the algorithm is also given. Some extensions are discussed.

Item ID: 53859 http://oa.upm.es/53859/ oai:oa.upm.es:53859 10.1016/j.spa.2018.11.017 Dr. Luis Mendo 07 Oct 2019 13:00 07 Oct 2019 13:00

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