Martingales, Tauberian theorem, and strategies of gambling

Using the Tauberian theorem, we get an asymptotic relation between the tail of the distribution of the quadratic characteristic of a martingale and the expectation of its terminal value. In case of continuous martingales the following result is proven: if $\tau $ is a stopping time for a standard Wiener process Wt with integrable terminal value $W_\tau $, then
\begin{equation} \liminf_{t\rightarrow \infty}(\mathbb P\{\tau >t\}\sqrt{t}) \ge \sqrt{\frac 2\pi }|\mathbb E W_\tau | . \end{equation}
Using a related result for discrete time martingales, we study asymptotic characteristics of some strategies of gambling and, in particular, Oscar's strategy.