Recurrence properties of autoregressive processes with super-heavy-tailed innovations

Abstract

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

Perturbing transient random walk in a random environment with cookies of maximal strength

“…We have P 0 = Q. Assumption A includes by (15) (16). Hence, the proof is completed by combining Theorems 2.2, 2.3 and Proposition 3.8.…”

Recurrence and transience of contractive autoregressive processes and related Markov chains

“…To the best of our knowledge there is at present no complete classification in simple terms of recurrence versus transience of these processes although this problem has been investigated for several decades, see [Pak75], [Pak79], [Kel92, Part I], [GM00, p. 1196], [ZG04], [Bau13] and the review below.…”

“…The case where (5) fails is sometimes referred to as super-heavy tailed, see [ZG04]. Among the few works which deal with recurrence versus transience of AR(1) processes are the unpublished preprint [Kel92, Part I] and [ZG04]. (For some recent work with deals with super-heavy tailed innovations see [BI15].)…”

Functional limit theorems for divergent perpetuities in the contractive case

“…To the best of our knowledge, (a) functional limit theorems for divergent perpetuities have not been obtained so far; (b) [13] is the only contribution to case (1.6) which deals with one-dimensional convergence. We would like to stress that outside the area of limit theorems we are only aware of two papers [12] and [19] which investigate case (1.6). Unlike (1.6) the critical non-contractive case E log |M | = 0 has received more attention in the literature, see [2,4,5,6,9,10,15].…”