  For a stochastic difference equation $D_n=A_nD_{n-1}+B_n$ which stabilises
upon time we study tail distribution asymptotics of $D_n$ under the assumption
that the distribution of $\log(1+|A_1|+|B_1|)$ is heavy-tailed, that is, all
its positive exponential moments are infinite. The aim of the present paper is
three-fold. Firstly, we identify the asymptotic behaviour not only of the
stationary tail distribution but also of $D_n$. Secondly, we solve the problem
in the general setting when $A$ takes both positive and negative values.
Thirdly, we get rid of auxiliary conditions like finiteness of higher moments
used in the literature before.
