Simulating a coin with irrational bias using rational arithmetic

TL;DR

This paper introduces an algorithm that simulates a Bernoulli random variable with an irrational parameter using only rational arithmetic and a series representation, solving an open problem for specific irrational values like Euler's constant.

Contribution

It provides a novel algorithm that efficiently simulates irrational Bernoulli parameters with rational arithmetic, including for Euler's constant, with bounded expected inputs and operations.

Findings

Expected number of inputs is at most 3.

Number of operations has a tail bounded by truncation error.

Successfully simulates Bernoulli with irrational parameters like Euler's constant.

Abstract

An algorithm is presented that, taking a sequence of independent Bernoulli random variables with parameter $1/2$ as inputs and using only rational arithmetic, simulates a Bernoulli random variable with possibly irrational parameter $\tau$. It requires a series representation of $\tau$ with positive, rational terms, and a rational bound on its truncation error that converges to $0$. The number of required inputs has an exponentially bounded tail, and its mean is at most $3$. The number of arithmetic operations has a tail that can be bounded in terms of the sequence of truncation error bounds. The algorithm is applied to two specific values of $\tau$, including Euler's constant, for which obtaining a simple simulation algorithm was an open problem.