Necessary and sufficient summability for almost sure convergence of stochastic difference equations

Let $(\xi_n)_{n\geq 1}$ be i.i.d. random variables satisfying

$$\mathop{\mathbf{E}}\ln(1+\xi_n)<0$$

and let $\alpha>0$ satisfy

$$\mathop{\mathbf{E}}(1+\xi_n)^\alpha-1=0.$$

For a given sequence $(S_n)_{n\geq 0}$, consider the stochastic difference equation

$$X_{n+1}=X_n(1+\xi_{n+1})+S_n,\qquad n\in\mathbb{N}_0.$$

Here, $S_n$ is $\alpha$-summable when it has the summability property defined in the preceding discussion of the paper. Necessary and sufficient summability conjecture. The solutions converge to zero almost surely if and only if $S_n$ is $\alpha$-summable. This conjecture proposes that the sufficient condition proved earlier is also necessary; the surrounding discussion points to a limsup counterexample as motivation, while no resolution is supplied in the given text.