Perpetuities with light tails and the local dependence measure

Abstract

This work investigates the tail behavior of solutions to the affine stochastic fixed-point equation of the form $X\stackrel{d}{=}AX+B$, where $X$ and $(A,B)$ are independent. Focusing on the light-tail regime, following [Burdzy et al. (2022), Ann. Appl. Probab.] we introduce a local dependence measure along with an associated Legendre-type transform. These tools allow us to effectively describe the logarithmic right-tail asymptotics of the solution $X$. Moreover, we extend our analysis to a related recursive sequence $X_n=A_n X_{n-1}+B_n$, where $(A_n,B_n)_{n}$ are i.i.d. copies of $(A,B)$. For this sequence, we construct deterministic scaling $(f_n)_{n}$ such that $\limsup_{n\to\infty} X_n/ f_n$ is a.s. positive and finite, with its non-random explicit value provided.