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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
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