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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 theleft tail of $X$ which complement well-known tail results of Kesten and Goldie.We generalize our results to ARMA processes with nonnegative coefficients whosenoise terms are from the IED class. We describe the lower envelope for theseARMA processes.

Inverse Exponential Decay: Stochastic Fixed Point Equation and ARMA Models