# Iterated random functions and regularly varying tails

## TL;DR

This paper studies the tail behavior of solutions to stochastic fixed point equations involving random Lipschitz functions, showing that the tail of the solution is comparable to that of the multiplicative coefficient under certain conditions.

## Contribution

It provides new insights into the tail asymptotics of solutions to stochastic fixed point equations with Lipschitz functions, extending results to cases approximated by linear functions.

## Key findings

- The tail of the solution R is comparable to the tail of A.
- The tail behavior of R depends on the tail of log(A).
- New results are obtained for the random difference equation.

## Abstract

We consider solutions to so-called stochastic fixed point equation $R \stackrel{d}{=} \Psi(R)$, where $\Psi $ is a random Lipschitz function and $R$ is a random variable independent of $\Psi$. Under the assumption that $\Psi$ can be approximated by the function $x \mapsto Ax+B$ we show that the tail of $R$ is comparable with the one of $A$, provided that the distribution of $\log (A\vee 1) $ is tail equivalent. In particular we obtain new results for the random difference equation.