Complexity of Anticipated Rejection Algorithms and the Darling-Mandelbrot Distribution

TL;DR

This paper analyzes the complexity of anticipated rejection algorithms, revealing that under certain conditions, their complexity converges to a Darling-Mandelbrot distribution, and provides explicit formulas and properties of this distribution.

Contribution

It establishes the limit law for the complexity of anticipated rejection algorithms as a Darling-Mandelbrot distribution and derives its explicit density and properties.

Findings

Complexity converges to Darling-Mandelbrot distribution under certain conditions.

Explicit density formula for the Darling-Mandelbrot distribution.

Analytic properties of the distribution are derived.

Abstract

We study in limit law the complexity of some anticipated rejection random sampling algorithms. We express this complexity in terms of a probabilistic process, the threshold sum process. We show that, under the right conditions, the complexity is linear and admits as a limit law a so-called Darling-Mandelbrot distribution, studied by Darling (Trans Am Math Soc 73:95-107, 1952) and Lew (Constr Approx 10(1):15-30, 1994). We also give an explicit form to the density of the Darling-Mandelbrot distribution and derive some of its analytic properties.