Square-root tail bound for martingale absorption ($\mathbb{P}(T>n)\le 2\tau /(\sigma \sqrt{n})$)

Under the conditional-variance and overshoot assumptions of the absorption-time bound from conditional variance and overshoot, $\mathbb{P}(T>n)\le 2\tau /(\sigma \sqrt{n})$. Bound the probability by $\mathbb{P}(T_R<\infty )+\mathbb{E}(T\wedge T_R)/n$ and optimize $1/R+{\tau }^{2}R/({\sigma }^{2}n)$. If the resulting bound is below one, its minimizing threshold is greater than one; otherwise the probability bound is trivial.