Practical Boundary Degeneracy and Reverse-Martingale Limits in Sequential Binary Models

A run of all failures, a run of all successes, or complete separation in a logistic regression each tempts the analyst to declare a probability of exactly zero or one. The central message of this paper is that all three phenomena share a common structure: finite sequential data justify practical boundary statements of the form $p\le \epsilon$ or $p\ge 1-\epsilon$, but not exact boundary probabilities. The paper’s contribution is to unify these three settings under a single reverse-martingale framework and to derive a stopping rule, ${\tau }_{\mathrm{R}\mathrm{M}}$, that requires three conditions simultaneously – boundary closeness $B_n\le \epsilon$, uncertainty width $W_n\le w$, and trajectory stability $r_n\le \eta$ – rather than boundary closeness alone. The reverse-martingale view recasts boundary degeneracy as a property of the limiting conditional law $M_{\mathrm{\infty }}=\text{E}(Y\text{given}{\text{G}}_{\mathrm{\infty }})$ rather than a finite-sample event, complementing classical one-sided binomial tests and Wald’s sequential probability ratio test without replacing them. Numerical studies across Bernoulli rare-event trials, low- and high-dimensional logistic regression, controlled risk trajectories, and a real health-economics data set demonstrate that boundary closeness alone is an unreliable stopping signal, and that the stability condition separates transient apparent certainty from genuine limiting degeneracy.