Fast Perfect Simulation of Vervaat Perpetutities

TL;DR

This paper introduces a significantly faster exact simulation method for Vervaat perpetuities, reducing the computational complexity from exponential to near-linear in the parameter , by extending coupling techniques to non-Markovian functions.

Contribution

The authors develop a novel simulation algorithm for Vervaat perpetuities that bounds the process from both sides, achieving a substantial speedup over previous methods.

Findings

Simulation time reduced from exponential to near-linear in .

Method extends coupling from the past to non-Markovian functions.

Exact sampling from Vervaat perpetuities becomes computationally feasible for larger .

Abstract

This work presents a faster method of simulating exactly from a distribution known as a Vervaat perpetuity. A parameter of the Vervaat perpetuity is $\beta \in (0,\infty)$. An earlier method for simulating from this distributon ran in time $O((2.23\beta)^{\beta}).$ This earlier method utilized dominated coupling from the past that bounded a stochastic process for perpetuities from above. By extending to non-Markovian update functions, it is possible to create a new method that bounds the perpetuities from both above and below. This new approach is shown to run in $O(\beta \ln(\beta))$ time.