Exit Time Analysis For Kesten's Stochastic Recurrence Equations
Chang-Han Rhee, Jeeho Ryu, Insuk Seo
TL;DR
This paper analyzes exit times for Kesten's stochastic recurrence equation X_{n+1}=A_{n+1}X_n+B_{n+1} in R^d, uncovering a sharp dichotomy governed by the sign of the Lyapunov exponent γ_L.In the contractive regime (γ_L<0) the mean exit time E[τ_R(x0)] grows polynomially with radius R, with exponent α solving h_A(α)=1, and α coinciding with the tail index of the stationary distribution; the authors provide both upper and lower bounds and a univariate refinement when α≥2.In the explosive regime (γ_L>0) the exit time scales logarithmically with R: E[τ_R(x0)] is sandwiched between constants times log R, with a nearly sure lower bound of 1/γ_L and a technically involved upper bound derived from a submartingale construction.The results hold under a broad yet verifiable set of assumptions that are weaker than classical Kesten conditions and are illustrated via applications to stochastic gradient descent on quadratic losses and ARCH/GARCH time-series models, solidifying the link between stochastic recurrence dynamics and practical algorithms.
Abstract
Kesten's stochastic recurrent equation is a classical subject of research in probability theory and its applications. Recently, it has garnered attention as a model for stochastic gradient descent with a quadratic objective function and the emergence of heavy-tailed dynamics in machine learning. This context calls for analysis of its asymptotic behavior under both negative and positive Lyapunov exponents. This paper studies the exit times of the Kesten's stochastic recurrence equation in both cases. Depending on the sign of Lyapunov exponent, the exit time scales either polynomially or logarithmically as the radius of the exit boundary increases.