Inverse exponential decay: Stochastic fixed point equation and ARMA models

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” ($\operatorname{IED}$) 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 $\operatorname{IED}$ class. We describe the lower envelope for these ARMA processes.