Approximating Perpetuities

TL;DR

This paper introduces an improved algorithm for approximating the distribution functions and densities of perpetuities, with applications to Quickselect, offering better efficiency and practical accuracy.

Contribution

It refines a previous discretization-based method, reducing complexity and extending effective approximation to densities with promising experimental results.

Findings

Algorithm significantly reduces computational complexity.

Distribution functions are approximated accurately.

Densities can also be approximated well despite weaker theoretical bounds.

Abstract

We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be approximated well.