Appendix to "Approximating perpetuities"

TL;DR

This paper presents a simple, four-line algorithm for perfect simulation of a specific distribution related to Quickselect analysis, using coupling from the past and a multigamma coupler.

Contribution

It introduces a novel, concise simulation method for a distribution defined by a fixed point equation, applicable in probabilistic analysis of algorithms.

Findings

Efficient perfect simulation algorithm with minimal code

Applicable to distribution from a fixed point equation in Quickselect analysis

Demonstrates practical implementation of coupling from the past with a multigamma coupler.

Abstract

An algorithm for perfect simulation from the unique solution of the distributional fixed point equation $Y=_d UY + U(1-U)$ is constructed, where $Y$ and $U$ are independent and $U$ is uniformly distributed on $[0,1]$. This distribution comes up as a limit distribution in the probabilistic analysis of the Quickselect algorithm. Our simulation algorithm is based on coupling from the past with a multigamma coupler. It has four lines of code.