# Stochastic recursions: between Kesten's and Grincevi\v{c}ius-Grey's assumptions

## TL;DR

This paper analyzes the tail behavior of solutions to stochastic recursions with Lipschitz functions near affine transformations, focusing on cases where the multiplicative component's moments are critical and the additive noise has heavy tails.

## Contribution

It characterizes the tail asymptotics of stationary solutions under critical moment conditions and extends second order asymptotics for affine recursions.

## Key findings

- Tail of stationary solution follows a power law with index -α.
- Second order asymptotics are derived for affine recursions.
- Results connect Kesten's and Grincevičius-Grey's assumptions.

## Abstract

We study the stochastic recursion $X_n=\Psi_n(X_{n-1})$, where $(\Psi_n)_{n\geq 1}$ is a sequence of i.i.d. random Lipschitz mappings close to the random affine transformation $x\mapsto Ax+B$. We describe the tail behaviour of the stationary solution $X$ under the assumption that there exists $\alpha>0$ such that $\mathbb{E} |A|^{\alpha}=1$ and the tail of $B$ is regularly varying with index $-\alpha<0$. We also find the second order asymptotics of the tail of $X$ when $\Psi(x)=Ax+B$.