Sharp variance and functional limits for thinned counts in infinite birth–death chains

For nonexplosive irreducible positive-recurrent continuous-time birth–death chains, a cumulative edge-energy criterion exactly characterizes finite stationary asymptotic variance of independently retained jump counts. Failure of the criterion implies variance divided by time tends to infinity. Finite energy yields a functional central limit theorem from every fixed initial law, while variance convergence is established under bounded stationary density. Only finite mean retained intensity is required, allowing unbounded intensities, non-square-integrable correctors and infinite mean raw jump activity. An extension gives explicit joint covariances for several count streams with correlated same-jump marks. Written infinite-state arguments are distinguished from exact finite-generator checks. This is an unrefereed research candidate, not proof-assistant formalization or external peer review.