The plug-in empirical rule for the last-success problem achieves asymptotic oracle optimality when p_n -> 0 with n p_n -> infinity and finite-horizon bounds for p >= p0 > 0, but no oracle-free rule converges uniformly to the oracle value over all p in (0,1).
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Stopping on the last success with unknown odds: Impossibility barriers and quantitative oracle bounds
The plug-in empirical rule for the last-success problem achieves asymptotic oracle optimality when p_n -> 0 with n p_n -> infinity and finite-horizon bounds for p >= p0 > 0, but no oracle-free rule converges uniformly to the oracle value over all p in (0,1).