This lemma can be applied e.g.~in the settings of a recent paper of Kevei . The
random difference equation (1.1) is considered there in two cases: (1) if $E A^\alpha
= 1$ for some $\alpha > 0$, but $E A^\alpha log_+ A = \infty$; (2) there is $\alpha
> 0$ such that $ E A^\alpha < 1$, but $E A^s = \infty$ for all $s > \alpha$.
Then, under some more detailed assumptions, applying the renewal type argument, Kevei
 proved analogous results to Theorem 1.3. Of course with a slightly different
Dariusz Buraczewski. A simple proof of heavy tail estimates for affine type Lipschitz
recursions. (2016) STOCHASTIC PROCESSES AND THEIR APPLICATIONS 0304-4149 126