Recurrence and transience of random difference equations in the critical case

For i.i.d. random vectors $(M_1,Q_1),(M_2,Q_2),\dots$ such that $M>0$ a.s., $Q\ge 0$ a.s. and $\mathbb{P}(Q=0)<1$, the random difference equation $X_n=M_n{X}_{n-1}+Q_n$, $n=1,2,\dots$, is studied in the critical case when the random walk with increments $\log M_1,\log M_2$ is oscillating. We provide conditions for the null-recurrence and transience of the Markov chain $(X_n{)}_{n\ge 0}$ by inter alia drawing on techniques developed in the related article Alsmeyer et al (2017) for another case exhibiting the null-recurrence/transience dichotomy.