Exit Time Analysis For Kesten's Stochastic Recurrence Equations

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.