Sum the Probabilities to \(m\) and Stop

This work investigates the optimal selection of the \(m\)th last success in a sequence of \(n\) independent Bernoulli trials. We propose a threshold strategy that is \(\varepsilon\)-optimal under minimal assumptions about the monotonicity of the trials' success probabilities. This new strategy ensures stopping at most one step earlier than the optimal rule. Specifically, the new threshold coincides with the point where the sum of success probabilities in the remaining trials equals \(m\). We show that the underperformance of the new rule, in comparison to the optimal one, is of the order \(O(n^{-2})\) in the case of a Karamata-Stirling success profile with parameter \(\theta > 0\) where \(p_k = \theta / (\theta + k - 1)\) for the \(k\)th trial. We further leverage the classical weak convergence of the number of successes in the trials to a Poisson random variable to derive the asymptotic solution of the stopping problem. Finally, we present illustrative examples highlighting the close performance between the two rules.