On Densities for Solutions to Stochastic Fixed Point Equations

TL;DR

This paper proves the existence of smooth, rapidly decreasing density functions for solutions to stochastic fixed-point equations, which are key in analyzing recursive algorithms and structures.

Contribution

It extends the analysis of stochastic fixed-point equations by establishing the existence of well-behaved density functions for their solutions.

Findings

Existence of bounded, smooth densities for solutions

Applicable to limits in recursive algorithms like Quicksort

Enhances understanding of asymptotic behaviors

Abstract

We consider systems of stochastic fixed-point equations that arise in the asymptotic analysis of random recursive structures and algorithms such as Quicksort, generalized P\'olya urn processes and path lengths of random recursive trees and split trees. Based on an approach of Fill and Janson for the analysis of the Quicksort-limit, the main result of this paper is the existence of bounded, smooth, rapidly decreasing density functions for limits given by these kinds of limit equations.