I am dealing with a positively bounded martingale sequence generated by the process
$Y_t = \begin{cases} 3Y_{t-1}/Y_0, \quad \ \text{with probability} \ \frac{1}{4} \\ \frac{1}{3}Y_{t-1}/Y_0, \quad \text{with probability} \ \frac{3}{4} \end{cases} \quad t \in \mathbb{N}$
for some deterministic $Y_0 > 0$. From simulation i guess that $Y_t \xrightarrow[]{n \rightarrow \infty}0$ but i cannot come up with a way to show this explicitly.