I have a discrete process $X_t,~t\in \mathbb N$ for which I have shown that $$\mathbb E[X_t|\mathcal F_{t-1}] \geq \alpha X_{t-1}^2$$ for some positive constant $\alpha \in (0,0.5)$. Also, $X_t$'s take values in $[0,1]$.
If this had been a submartingale then I'd conveniently use Doob's inequality on it to get $\Pr(\sup_{t\in 1..T}X_t \geq \epsilon) \leq \frac{\mathbb E[X_T]}{\epsilon}$. But this is not the case, and I'm wondering if an inequality in the same spirit exists which I could use to bound $\Pr(X_t \geq \epsilon)$ ... (I don't care for the inequality to be uniform over time)
Many Thanks!