Heterogeneous Cutoffs in Stopped Bernoulli Trials: Sharp Thresholds and Finite-Sample Bias

We consider m independent components in which Bernoulli trials have success probability p. In component i, trials stop at the first success or upon reaching the prescribed cutoff ni, whichever occurs first. We study how the expected proportion of successes among all trials performed changes as the cutoff vector is enlarged. Along the homogeneous cutoff sequence n1 = ··· = nm = n, the expectation increases strictly at every step. For a single-coordinate step a → a + 1, strict increase uniformly over all finite backgrounds holds if and only if p ≥ 2/(a + 2), and universal coordinate-wise monotonicity therefore has the sharp threshold p = 2/3. The same thresholds govern simultaneous unit increases of all coordinates of finite cutoff vectors. For every p, monotonicity is restored whenever all remaining cutoffs are at most a + 1, yielding a canonical two-level bridge between successive homogeneous cutoff vectors. The same argument further implies that the expectation is strictly greater than p for every cutoff configuration other than the all-one configuration, including those with unrestricted coordinates. 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. Below this threshold, an all-finite configuration can exceed the unrestricted expectation, although the corresponding expectations still converge to the unrestricted expectation as all cutoffs tend to infinity.