Persistence of autoregressive sequences with logarithmic tails

Abstract

We consider autoregressive sequences Xn = aXn−1 + ξn and Mn = max{aMn−1, ξn} with a constant a ∈ (0, 1) and with positive, independent and identically distributed innovations {ξk}. It is known that if P(ξ1 > x) ∼ d log x with some d ∈ (0,− log a) then the chains {Xn} and {Mn} 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 {Xn} and {Mn} conditioned to stay over a fixed level x0. Furthermore, we study tail asymptotics for recurrence times of {Xn} and {Mn} in the case when these chains are positive recurrent and the tail of log ξ1 is subexponential.