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Set $G_n=U_n-\theta V_n$, and define the estimator on zero-denominator events as part of the model. On $V_n>\delta >0$, its error is at most $|G_n|/\delta$. Centering $G_n$ and applying the Cauchy-Schwarz inequality gives the displayed bound, where $r_n$ is the expected error on $V_n\le \delta$. This controls a ratio estimator without falsely assuming that its numerator and denominator are independent.

A closed-population capture-recapture model assumes the same individuals remain throughout the observations: no births, deaths, immigration, or emigration. Its elementary two-sample version additionally requires retained, correctly recognized marks and a second sample chosen by simple random sampling independently of first-sample membership. Mixing and unchanged capture behavior help make this plausible. Equal individual capture probabilities alone do not guarantee that every size-$C$ subset is equally likely; that stronger sampling assumption gives the hypergeometric distribution. See Barker, §§1.1.1 and 1.1.3: closure, marks, and catchability assumptions.

Unequal catchability, mark loss, and population change can invalidate this capture-recapture model. For example, if especially catchable individuals appear repeatedly, overlap can be inflated and the Lincoln-Petersen estimator can underestimate population cardinality. Lost marks can decrease recorded overlap and increase the same ratio estimator, all else equal. Two samples alone cannot establish that the sampling assumptions hold.

The Lincoln-Petersen estimator is the ratio estimator obtained by matching the observed overlap to its hypergeometric expected value. It is defined when $R>0$. To distinguish that ratio from the exact integer maximization, consecutive positive likelihoods satisfyThis comparison is valid for $N\ge M+C-R$. For the toy counts, the numerator minus the denominator is $500-5(N+1)$. Thus the likelihood function increases up to ninety-nine, ties at ninety-nine and one hundred, then decreases for every larger integer. This independently establishes the global maximum-likelihood estimates, beyond the plotting window. The adjacent-ratio method is given in Guiteras, §8.