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For fixed $\theta >0$, a geometric mutation count in a coalescent epoch with mean ${\mu }_i=\theta /i$ has third moment ${\mu }_i+6{\mu }_i^2+6{\mu }_i^3$. The centered absolute third moment is at most $4\mathbb{E}[Y_i^3]+4{\mu }_i^3$, so its sum is $O(\log n)$. The total variance is asymptotic to $\theta \log n$, making the Lyapunov condition with $\delta =1$ hold. Rescaling by the harmonic sum gives the displayed limit. At $\theta =0$ the estimator is identically zero and the limit is degenerate. The variance of the Watterson estimator decreases only logarithmically with sample size at a fixed nonrecombining locus.

The Watterson estimator isThe expectation above makes it an unbiased estimator. Its variance, given by the variance of the Watterson estimator, isIndeed, the harmonic sum satisfies $H_{n-1}\sim \log n$ and $H_{n-1}^{(2)}$ stays bounded. This proves mean-square convergence; the Chebyshev inequality then shows that it is a consistent estimator, since $P(|\hat{\theta }-\theta |>\epsilon )\le \mathrm{Var}(\hat{\theta })/{\epsilon }^2\to 0$.