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

Abstract

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, $\ddot{I}{„}_{\mathrm{RM}}$, that requires three conditions simultaneously -- boundary closeness $B_n\le \epsilon$, uncertainty width $W_n\le w$, and trajectory stability $r_n\le \hat{I}\cdot$ -- rather than boundary closeness alone. The reverse-martingale view recasts boundary degeneracy as a property of the limiting conditional law $M_\infty=\E(Y\given\G_\infty)$ 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.

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