Extended One-Liners for the Beta, Gamma, and Dirichlet Distributions with Shape Parameters Below One

Abstract

We present an explicit deterministic transformation of a fixed number of i.i.d. uniform random variables with exact Beta$(a,1-a)$ law for $0<a<1$, using only elementary operations (an ``extended one-liner'', see \cite{devroye1996oneline}). As corollaries, the families Beta$(a,b)$ with $\min(a,b)<1$, Gamma$(c)$ with $c<1$, and Dirichlet$(\alpha_1,\dots,\alpha_d)$ with $0<\alpha_i<1$, for fixed $d$, also have extended one-\liners.