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The asymptotic normality of the Watterson estimator gives approximate standard error $\sqrt{\theta /\log n}$. Thus precision at a fixed nonrecombining locus improves only logarithmically with sample size. To halve the asymptotic standard error, one needs $\log n^{\prime }\simeq 4\log n$, or roughly $n^{\prime }=n^4$, rather than four times as many sampled chromosomes.

For large samples and positive $\hat{\theta }$, a plug-in normal approximation gives $\hat{\theta }\pm 1.96\sqrt{\hat{\theta }/\log n}$ as an approximate 95% confidence interval. At low mutation counts it can perform poorly or give a negative endpoint; the exact count model is then preferable. Multiple independent, comparable loci can provide additional independent information, whereas tightly linked chromosomes still share the random ancestry at one locus. Genetic recombination, nonneutral evolution and changing population size require corresponding changes to the model.

For fixed $\theta >0$, use the Lyapunov condition for the independent centered epoch counts $Z_i=Y_{i+1}-\theta /i$. Their total variance is $s_n^2=\theta H_{n-1}+{\theta }^2{H}_{n-1}^{(2)}\sim \theta \log n$.

Write ${\mu }_i=\theta /i$. Differentiating the geometric distribution probability generating function gives $\mathbb{E}[Y_{i+1}^3]={\mu }_i+6{\mu }_i^2+6{\mu }_i^3$. Since $|Y_{i+1}-{\mu }_i{|}^3\le 4Y_{i+1}^3+4{\mu }_i^3$, the sum of the centered absolute third moments is $O(\log n)$. ConsequentlyThe Lindeberg-Feller central limit theorem applies through the Lyapunov condition, giving $(S-\theta H_{n-1})/s_n\stackrel{d}{\to }N(0,1)$. Now $\sqrt{\log n}{s}_n/H_{n-1}\to \sqrt{\theta }$, so the Slutsky theorem proves the asymptotic normality of the Watterson estimator:The limiting variance is $\theta$. When $\theta =0$, every count vanishes and the limiting law is the point mass at zero.