  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.
