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

Summary

A detector may record only a fraction of the jumps of a random process. Even when the average detected count is finite, its long-run uncertainty can be enormous. This paper identifies an exact boundary for a broad class of infinite birth–death chains: one cumulative edge-energy series determines whether the stationary variance has a finite growth rate.

Finite energy also gives a Brownian functional limit. But a Brownian limit is not a promise that variances behave well for every starting distribution. A worked counterexample separates these two conclusions. The paper further handles several detectors, including correlated detection decisions at the same jump.

Summary for specialists

Let $c_k=\pi_k\lambda_k$ be the edge conductance of a nonexplosive, irreducible, positive recurrent continuous-time birth–death chain. Upward and downward retentions have probabilities $a_k,b_k$, with $0<r=\sum_k c_k(a_k+b_k)<\infty$. Put $\delta_k=(a_k-b_k)/2$, let $v$ be the symmetrised retained intensity, and define $H_k=\sum_{i\le k}\pi_i(v_i-r)$.

The extended stationary variance rate is

$$\sigma^2=r-2\sum_k c_k\delta_k^2+2\sum_k\frac{H_k^2}{c_k}.$$

It is finite exactly when the last series is finite. In that case the centred count has a functional central limit theorem from every fixed initial probability law. Convergence of variance is established under stationarity and under initial laws with bounded density relative to $\pi$, not for arbitrary initial laws.

Technical account

The proof decomposes the count into a symmetric edge contribution and a directional endpoint correction. Detailed balance turns the ordered pair-of-jumps calculation into the covariance kernel of the symmetrised intensity. An $L^1$ resolvent argument identifies its integrated value with the cumulative edge energy. A forward–backward martingale estimate controls an unbounded corrector by its gradient energy rather than by an unnecessary square-integrability assumption on the corrector itself.

The joint-count extension permits dependence between coordinates of a mark at the same jump, while marks at distinct jumps remain conditionally independent given the chain path. Its covariance matrix contains a shared-chain energy term and a simultaneous-mark term. Consequently, identical detectors and conditionally independent detectors can have the same marginal retention probabilities but different cross-covariances.

Evidence, assurance and limitations

The infinite-state conclusions are analytic theorems in the manuscript. Exact finite-generator, renewal, arithmetic and joint-mark checks test identities and normalisations; they are not a proof-assistant verification or an unaffiliated replication of the analytic argument. The candidate is unrefereed and internally assessed.

Finite retained intensity is required, but finite mean raw activity and square-integrable retained intensity are not. Divergent edge energy establishes divergent stationary variance per unit time; it does not rule out every possible weak limit or alternative scaling. The result concerns a line-shaped birth–death graph, not arbitrary networks.

Relationship to earlier work

Daley, Nazarathy and Wang supply the finite-state thinned-count setting. Chojecki's 2011 theorem is an important close antecedent: it uses a cumulative-square criterion for a discrete-time state observable. General jump-count functional limits and reversible additive-functional methods are also established. The manuscript distinguishes those results from its continuous-time marked-count theorem under the stated weaker integrability conditions and its explicit variance alternative.

Who should care, and why

Why the problem matters

A mean count can look ordinary while its uncertainty is pathological. A sharp criterion tells the modeller exactly which infinite tail must be controlled, and why finite-state truncations alone cannot settle the question.

How to inspect or reproduce the recorded checks

Read the notation guide and main theorem first, then follow the energy, count-variance and corrector lemmas. The package README and AI index map each claim to its proof and replay files. Reproduce the finite-generator and renewal identities, inspect the infinite-state examples analytically, and run the joint-mark covariance checks separately. Finite tests support the formulas but do not replace the infinite-state proof.

The most valuable next projects

Develop certified tail bounds for usable finite truncations, characterise non-Gaussian scaling when the energy diverges, or investigate which parts of the explicit criterion survive on graphs with branches. Each requires new arguments beyond the published theorem.

What is in the evidence package

A self-contained manuscript and PDF, exact finite-state and example calculations, joint-mark checks, replay instructions and environment information, a claim-to-evidence index, and a source comparison preserving the boundary between established methods and the candidate's contribution.

@misc{infinitebirthdeaththinnedcounts2026,
  title        = {Sharp variance and functional limits for thinned counts in infinite birth–death chains},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.23448519},
  url          = {https://doi.org/10.5281/zenodo.23448519},
  version      = {1.0.0},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/infinite-birth-death-thinned-counts/}
}