Persistence of autoregressive sequences with logarithmic tails

TL;DR

This paper analyzes the tail behavior of recurrence times in autoregressive sequences with logarithmic tail innovations, revealing their regular variation and limit laws under different recurrence regimes.

Contribution

It provides new results on the tail asymptotics and limit theorems for autoregressive sequences with logarithmic tail innovations, extending understanding of their recurrence properties.

Findings

Recurrence times have regularly varying tails with index -1-d/log a.

Limit theorems are established for sequences conditioned to stay above a level.

Tail asymptotics are derived for positive recurrent chains with subexponential tails.

Abstract

We consider autoregressive sequences $X_n=aX_{n-1}+\xi_n$ and $M_n=\max\{aM_{n-1},\xi_n\}$ with a constant $a\in(0,1)$ and with positive, independent and identically distributed innovations $\{\xi_k\}$. It is known that if $\mathbf P(\xi_1>x)\sim\frac{d}{\log x}$ with some $d\in(0,-\log a)$ then the chains $\{X_n\}$ and $\{M_n\}$ are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index $-1-d/\log a$. We also prove limit theorems for $\{X_n\}$ and $\{M_n\}$ conditioned to stay over a fixed level $x_0$. Furthermore, we study tail asymptotics for recurrence times of $\{X_n\}$ and $\{M_n\}$ in the case when these chains are positive recurrent and the tail of $\log\xi_1$ is subexponential.