Aggregate Stability in Repeated Restricted Geometric Experiments with Heterogeneous Cutoffs

Consider m independent Bernoulli sequences with a common success probability 0 < p < 1, each stopped at the first success or at a prescribed cutoff. We allow the cutoffs to form an arbitrary deterministic sequence, with finite and unrestricted cutoffs permitted. We prove that the aggregate success rate converges almost surely to p as m → ∞, and that the corresponding expectations also converge to p. Previous work has shown that, at every finite stage, the expected aggregate success rate is strictly greater than p whenever the cutoff vector is not the all-one vector; two direct derivations of this finite-stage inequality, one analytic and one probabilistic, are included here for completeness. Nevertheless, cutoff configurations that affect the expected aggregate rate at finite stages do not affect its limiting value.