# Reverse Hoeffding Inequalities

Suppose that $$X_t$$ is a super-martingale, the Hoeffding inquality gives an exponential upper bound on the quatity $$\mathbb{P}\left( \sup_{0 \leq t\leq T}X_t \geq x \right).$$

When can a lower-bound be obtained; that is a function of $$x$$ and $$t$$ such that $$f(t,x)\leq \mathbb{P}\left( \sup_{0 \leq t\leq T} X_t \geq x \right)?$$