Homogeneous Cutoff Monotonicity and a Sharp 2/3 Coordinate-Wise Threshold in Repeated Restricted Geometric Experiments

Abstract

We consider m independent sequences of Bernoulli trials with success probability p, each terminated at its first success or after n trials, whichever occurs first. We study how the expected proportion of successes among all trials performed across the m components changes as the cutoff parameters are enlarged. Along the homogeneous cutoff sequence, the expectation increases strictly at every step for all m, n ≥ 1 and 0 < p < 1. Allowing the component cutoffs to differ reveals a different structure. For a fixed cutoff step a → a + 1, the expectation increases strictly for every finite choice of the remaining cutoffs if and only if p ≥ 2/(a + 2). Hence coordinate-wise monotonicity for all cutoff steps and finite heterogeneous backgrounds holds if and only if p ≥ 2/3. Nevertheless, monotonicity is restored for every p whenever all remaining cutoffs are at most a + 1. In particular, the expectation increases strictly at each coordinate step along the canonical two-level bridge from (n, ..., n) to (n + 1, ..., n + 1). We also compare restricted and unrestricted configurations. Along the homogeneous cutoff sequence, expectations increase to the unrestricted expectation from below. Allowing unrestricted components in the heterogeneous model, we prove that the fully unrestricted expectation is a strict upper bound for every configuration with at least one finite cutoff if and only if p ≥ 1/2. When p < 1/2, even the expectation for an all-finite configuration can exceed the fully unrestricted expectation, although convergence to the unrestricted expectation still holds as all cutoffs tend to infinity. Finally, an exact counterexample shows that successive homogeneous cutoff increments need not decrease, so first-order cutoff monotonicity does not extend to the corresponding second-order property.