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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 $$ f(t,x)\leq \mathbb{P}\left( \sup_{0 \leq t\leq T} X_t \geq x \right)? $$

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