On invariant measures of stochastic recursions in a critical case

TL;DR

This paper investigates the behavior at infinity of the unique invariant measure for a critical autoregressive process with random coefficients, also exploring stationary measures of related stochastic recursions, including applications in queuing theory.

Contribution

It characterizes the asymptotic behavior of the invariant measure in the critical case and analyzes stationary measures of related stochastic recursions, extending previous results.

Findings

Describes the tail behavior of the invariant measure at infinity.

Provides insights into stationary measures in queuing models.

Extends understanding of critical stochastic recursions.

Abstract

We consider an autoregressive model on $\mathbb{R}$ defined by the recurrence equation $X_n=A_nX_{n-1}+B_n$, where $\{(B_n,A_n)\}$ are i.i.d. random variables valued in $\mathbb{R}\times\mathbb{R}^+$ and $\mathbb {E}[\log A_1]=0$ (critical case). It was proved by Babillot, Bougerol and Elie that there exists a unique invariant Radon measure of the process $\{X_n\}$. The aim of the paper is to investigate its behavior at infinity. We describe also stationary measures of two other stochastic recursions, including one arising in queuing theory.