Limiting distribution of harmonic sample means of i.i.d. $U(0,1)$s after appropriate scaling.

Question

Suppose $\{X_n\}$ is a sequence of i.i.d. random variables which follow the uniform distribution on $(0,1)$. Denote $$M_n=\frac{X_1+\cdots+X_n}{n},\;H_n=\frac{n}{\frac{1}{X_1}+\cdots+\frac{1}{X_n}}.$$ By Strong Law of Large Numbers we know $M_n\to\frac{1}{2},H_n\to0$ almost surely. Note for $p\geq1$, we have $$\mathbb{E}M_n^p\leq\mathbb{E}\frac{X_1^p+\cdots+X_n^p}{n}=\mathbb{E}X_1^p,$$ thus $M_n$ is bounded in $L^p$ for any $p\geq1$. Note $H_n\leq M_n$ by Cauchy-Schwarz inequality, we know $\{H_n,M_n|n\geq1\}$ is bounded in $L^p$. Hence we know for any $r\geq1$, we have $$\mathbb{E}M_n^r\to2^{-r},\mathbb{E}H_n^r\to0.$$ Now I want to find a sequence $\{a_n\}$ so that $a_nH_n$ converges (in distribution) to some nonzero distribution, but I don't know how to deal with it. Any help will be appreciated.

Answer 1

According to this paper, theorem 2.5, we have: $$H_n \cdot \ln^2(n)-\ln(n) \xrightarrow[n\to +\infty]{\mathcal D} X$$ where $X$ has the density function $f_X(x) = g(-x)$ with $g$ defined by the formula $(12)$.