= Asymptotic normality of the Watterson estimator
{title2=$\sqrt{\log n}(\widehat\theta_W-\theta)\xrightarrow{d}N(0,\theta)$}
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.
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