Convergence of the critical probabilities for stopping near the top of a random walk
Let \((S_n)_{n\ge0}\) be a Bernoulli random walk with \(P(X_i=1)=p\) and \(P(X_i=-1)=q=1-p\), and let \(M_n=\max\{S_0,\ldots,S_n\}\). We consider the semi-best selection problem of maximizing \(P(M_N-S_τ\le1)\) over stopping times \(τ\le N\). Allaart showed that the optimal decision in the critical state \((n,1)\) is determined by a sequence of critical probabilities \(\{p_n\}\), and conjectured that \(p_n\to1/2\). We prove this conjecture. For every fixed \(p<1/2\), the stopping value converges to \(ν=1-(p/q)^2\), while the continuation value converges to \(2pν\). Since \(2pν<ν\), stopping is optimal in state \((n,1)\) for all sufficiently large \(n\). Together with the known bound \(p_n<1/2\) for \(n\ge4\), this yields \(\lim_{n\to\infty}p_n=1/2\).