Slowly varying asymptotics for signed stochastic difference equations

TL;DR

This paper analyzes the tail behavior of solutions to a stochastic difference equation with heavy-tailed coefficients, extending results to cases with sign-changing coefficients and removing previous moment restrictions.

Contribution

It provides a comprehensive asymptotic analysis of the tail distribution for the process, including non-positive coefficients and without requiring higher moment conditions.

Findings

Characterizes tail asymptotics of the process D_n

Handles both positive and negative values of A

Eliminates the need for finiteness of higher moments

Abstract

For a stochastic difference equation $D_n=A_nD_{n-1}+B_n$ which stabilises upon time we study tail distribution asymptotics of $D_n$ under the assumption that the distribution of $\log(1+|A_1|+|B_1|)$ is heavy-tailed, that is, all its positive exponential moments are infinite. The aim of the present paper is three-fold. Firstly, we identify the asymptotic behaviour not only of the stationary tail distribution but also of $D_n$. Secondly, we solve the problem in the general setting when $A$ takes both positive and negative values. Thirdly, we get rid of auxiliary conditions like finiteness of higher moments used in the literature before.