= Solution
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 $s_n^2=\theta H_{n-1}+\theta^2H_{n-1}^{(2)}\sim\theta\log n$.
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 $|Y_{i+1}-\mu_i|^3\le4Y_{i+1}^3+4\mu_i^3$, the sum of the centered absolute third <moments> is $O(\log n)$. Consequently
$$
\frac{\sum_{i=1}^{n-1}\mathbb E|Z_i|^3}{s_n^3}=O((\log n)^{-1/2})\longrightarrow0.
$$
The <Lindeberg-Feller central limit theorem> applies through the <Lyapunov condition>, giving $(S-\theta H_{n-1})/s_n\xrightarrow{d}N(0,1)$. Now $\sqrt{\log n}\,s_n/H_{n-1}\to\sqrt\theta$, so the <Slutsky theorem> proves the <asymptotic normality of the Watterson estimator>:
$$
\boxed{\sqrt{\log n}(\widehat\theta-\theta)\xrightarrow{d}N(0,\theta).}
$$
The limiting <variance> is \b[$\theta$]. When $\theta=0$, every count vanishes and the limiting law is the point mass at zero.
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