PUB - Publikationen an der Universität Bielefeld

We consider autoregressive sequences X-n = aX(n-1) + xi(n) and M-n = max{aM(n-1), xi(n)} with a constant a is an element of (0, 1) and with positive, independent and identically distributed innovations {xi(k)}. It is known that if P(xi(1) > x) similar to d/log x with some d is an element of (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.