Inverse Exponential Decay: Stochastic Fixed Point Equation and ARMA Models

TL;DR

This paper investigates the tail behavior of solutions to a stochastic fixed point equation with inverse exponential decay noise and extends the analysis to ARMA processes, providing new tail results and describing their lower envelope.

Contribution

It introduces novel tail analysis for fixed point equations with IED noise and generalizes these results to ARMA models, including their lower envelope characterization.

Findings

Derived new tail asymptotics for solutions to the fixed point equation.

Extended tail results to ARMA processes with IED noise.

Characterized the lower envelope of ARMA processes with IED noise.

Abstract

We study solutions to the stochastic fixed point equation $X\stackrel{d}{=}AX+B$ when the coefficients are nonnegative and $B$ is an "inverse exponential decay" (IED) random variable. We provide theorems on the left tail of $X$ which complement well-known tail results of Kesten and Goldie. We generalize our results to ARMA processes with nonnegative coefficients whose noise terms are from the IED class. We describe the lower envelope for these ARMA processes.