Consider the problem of maximizing the probability of stopping with one of the two highest values in a Bernoulli random walk with arbitrary parameter $p$ and finite time horizon $n$. Allaart \cite{ Allaart} proved that the optimal strategy is determined by an interesting sequence of constants $\{ p_{ n}\}$. He conjectured the asymptotic behavior to be $1/2$. In this work the best lower bound for this sequence is found and more of its properties are proven towards solving the conjecture.
Categories: math.PR