Recurrence Properties of Autoregressive Processes With Super-Heavy Tailed Innovations

This paper studies recurrence properties of autoregressive (AR) processes with “super-heavy tailed” innovations. Specifically, we study the case where the innovations are distributed, roughly speaking, as log-Pareto random variables (i.e., the tail decay is essentially a logarithm raised to some power). We show that these processes exhibit interesting and somewhat surprising behavior. In particular, we show that AR(1) processes, with the usual root assumption that is necessary for stability, can exhibit null-recurrent as well as transient dynamics when the innovations follow a log-Cauchy type distribution. In this regime, the recurrence classification of the process depends, somewhat surprisingly, on the value of the constant pre-multiplier of this distribution. More generally, for log-Pareto innovations, we provide a positive recurrence/null recurrence/transience classification of the corresponding AR processes. Short Title: AR processes with super-heavy tails