Perpetuities with light tails and the local dependence measure
Julia Le Bihan, Bartosz Kołodziejek
TL;DR
This work analyzes the right-tail behavior of light-tailed perpetuities given by $X \stackrel{d}{=} AX+B$ with dependent $(A,B)$, introducing a local dependence measure $g$ and a Legendre-type transform $\phi_{\rho}$ to characterize tail decay. By coupling the affine fixed-point equation with the recursive sequence $X_n = A_nX_{n-1}+B_n$ and a regularly varying tail scale $f$, the authors derive precise logarithmic tail asymptotics $\mathbb{P}(X>t)$, identify the fixed-point tail parameter $\lambda^{*}$ as $\phi_{\rho}$'s unique fixed point, and establish an almost-sure upper envelope for $X_n$ via $f^{-1}(\log n)$. The framework accommodates dependent $(A,B)$, with PQD yielding explicit expressions, and the results extend prior left-tail analyses (BKT22) to right-tail behavior in light-tailed regimes. An explicit Example demonstrates the theory and shows that certain previous lower bounds can be nonoptimal, underscoring the practical impact for stochastic fixed-point models and related processes.
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.