Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 39 / 1 / v / 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 $s_n^2=\theta H_{n-1}+{\theta }^2{H}_{n-1}^{(2)}\sim \theta \log n$.

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.