Let $(X\_n)$ be a sequence of i.i.d. random variables uniformly distributed in $[0,1]$; and, for $n\geq 1$, set
$$
S\_n = \sum\_{k=1}^n \frac{1}{\sqrt{X\_k}}\,.
$$
It follows from the generalized central limit theorem (as in [1] and [2, Theorem 3.1]) that
$$
\frac{S\_n-2n}{\sqrt{n\ln n}}
$$
converges in law to a Gaussian distribution. However, in this case Berry—Esseen fails to give any convergence rate, as the summands are not square integrable.


Moreover, Hall's results for this sort of sum do not apply. (In [3], this would correspond to $\alpha=2$, while the results hold for $0<\alpha<2$.)



> 
> What is the rate of convergence of $\frac{S\_n-2n}{\sqrt{n\ln n}}$ to Gaussian?
> 
> 
> 




---


[1] Shapiro, Jesse M., *“Domains of Attraction for Reciprocals of Powers of Random Variables.”* SIAM Journal on Applied Mathematics, vol. 29, no. 4, 1975, pp. 734–739. JSTOR, [www.jstor.org/stable/2100234](https://www.jstor.org/stable/2100234).


[2] Christopher S. Withers and Saralees Nadarajah, *[Stable Laws for Sums of Reciprocals](http://www.maths.manchester.ac.uk/~goran/research-reports/psrr17-2011.pdf)*. December 2011.


[3] Hall, P. (1981), *On the Rate of Convergence to a Stable Law.* Journal of the London Mathematical Society, s2-23: 179-192. [doi:10.1112/jlms/s2-23.1.179](https://dx.doi.org/10.1112/jlms/s2-23.1.179)



Rate of convergence of $\frac{1}{\sqrt{n\ln n}}(\sum_{k=1}^n 1/\sqrt{X_k}-2n)$, $X_i$ i.i.d. uniform on $[0,1]$?
$\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}} 
\newcommand{\PP}{\operatorname{\mathsf P}}$ 


Let $V\_k:=1/\sqrt{X\_k}$, $b\_n:=\sqrt{n\ln n}$, 
\begin{equation\*}
 Z\_n:=\frac{S\_n-2n}{\sqrt{n\ln n}}=\frac1{b\_n}\sum\_1^n (V\_k-\E V\_1), 
\end{equation\*}
\begin{equation\*}
 \de\_n:=\sup\_{x\in\R}|\De\_n(x)|, 
\end{equation\*}
where 
\begin{equation\*}
 \De\_n:=F\_n-G,\quad 
F\_n(x):= P(Z\_n<x),\quad G(x):=P(Z<x), 
\end{equation\*}
and $Z\sim N(0,1)$. 


We shall show that for all large enough $n$ 
\begin{equation\*}
 \frac{\sqrt{\ln\ln n}}{\ln n}\ll\de\_n\ll\ep\_n:=\frac{\ln\ln n}{\ln n}; \tag{1}
\end{equation\*}
here everywhere, the constants associated with $\ll$, $\gg$, $O(\cdot)$ are universal. Thus, we have a rather tight bracketing of $\de\_n$. (Conjecture: $\de\_n\asymp\frac{\ln\ln n}{\ln n}$.)


Let $c$ denote various complex-valued expressions (possibly different even within the same formula) such that $|c|\ll1$. 


The pdf of $V\_k$ is $x\mapsto\frac2{x^3}\,I\{x>1\}$, where $I$ denotes the indicator. 
Note that $|e^{iu}-1-iu|\le2|u|$ and $|e^{iu}-1-iu+u^2/2|\le|u|^3/6$ for real $u$. 
So, for the characteristic function (c.f.) $f\_V$ of $V\_k$ and $|t|\le1$ we have 
\begin{multline\*}
 \frac12\,f\_V(t)=\int\_1^\infty\frac{e^{itx}}{x^3}\,dx
 =\int\_1^\infty\frac{1+itx}{x^3}\,dx
 -\int\_1^{1/|t|}\frac{t^2x^2/2}{x^3}\,dx \\ 
 +\int\_1^{1/|t|}\frac{e^{itx}-1-itx+t^2x^2/2}{x^3}\,dx
 +\int\_{1/|t|}^\infty\frac{e^{itx}-1-itx}{x^3}\,dx \\
 =\frac12+it-\frac{t^2}2\,\ln\frac1{|t|}
 +c\int\_1^{1/|t|}\frac{|t|^3x^3}{x^3}\,dx
 +c\int\_{1/|t|}^\infty\frac{|t|x}{x^3}\,dx \\
 =\frac12+it-\frac{t^2}2\,\ln\frac1{|t|}+ct^2. 
\end{multline\*}
So, for $|t|\le1$ we have $\ln f\_V(t)=2it-t^2\,\ln\frac1{|t|}+ct^2$ and hence for the characteristic function $f\_n:=f\_{Z\_n}$ of $Z\_n$ we have 
\begin{multline\*}
 \ln f\_n(t)=-i2nt/b\_n+n\ln f\_V(t/b\_n)
 =-\frac{t^2}{\ln n}\,\ln\frac{\sqrt{n\ln n}}{|t|}+c\frac{t^2}{\ln n} \\ 
 =-\frac{t^2}2-\frac{t^2}{\ln n}\,\Big(\frac12\,\ln\ln n-\ln|t|\Big)+c\frac{t^2}{\ln n} \\ 
 =-\frac{t^2}2+\frac{t^2}{\ln n}\,\ln|t|+c\frac{t^2}{\ln n}\,\ln\ln n \tag{2}
\end{multline\*}
for $|t|\le b\_n=\sqrt{n\ln n}$. So, with $\ep\_n$ as in (1), 
\begin{equation\*}
 \ln f\_n(t)= 
 \begin{cases}
 -\frac{t^2}2+c\ep\_n|t| & \text{ if }|t|\le1 \\ 
 -\frac{t^2}2+c\ep\_n t^2\ & \text{ if }1\le|t|\le\ln n, 
 \end{cases}
\end{equation\*}
whence 
\begin{multline\*}
 \int\_{|t|<\ln n}\frac{|f\_n(t)-e^{-t^2/2}|}{|t|}\,dt \\ 
 \le \int\_{|t|<1}\frac{|e^{c\ep\_n|t|}-1|}{|t|}\,dt
 +\int\_{\R}\frac{|e^{-(1-2c\ep\_n)t^2/2}-e^{-t^2/2}|}{|t|}\,dt \le c\ep\_n;
\end{multline\*}
the latter integral was bounded using the identity $\int\_0^\infty\frac{e^{-at^2/2}-e^{-t^2/2}}t\,dt=\ln(1/\sqrt a)$ for $a>0$. 
By the Esseen smoothing inequality (see e.g. [formula (6.4)](https://arxiv.org/pdf/0903.3081)),
\begin{equation\*}
 \de\_n\le c\int\_{|t|<\ln n}\frac{|f\_n(t)-e^{-t^2/2}|}{|t|}\,dt +c/\ln n. 
\end{equation\*}
Now the second inequality in (1) immediately follows. 


It remains to prove the first inequality in (1). For real $t$ and real $A>0$, 
\begin{multline\*}
 \int\_0^\infty e^{itx}d\De\_n(x)=\int\_0^A e^{itx}d\De\_n(x)+c(1-F\_n(A))+c(1-G(A)) \\ 
 =c\De\_n(A)+c\De\_n(0)-it\int\_0^A e^{itx}\De\_n(x)dx+2c(1-G(A))+c\de\_n \\ 
 =c\de\_n+c|t|A\de\_n+ce^{-A^2/2} 
 =c\de\_n+c|t|\de\_n\sqrt{\ln\frac1{\de\_n}} 
\end{multline\*}
if $A=\sqrt{2\ln\frac1{\de\_n}}$. 
Similarly estimating $\int\_{-\infty}^0 e^{itx}d\De\_n(x)$, we have 
\begin{equation\*}
 f\_n(t)-e^{-t^2/2}=\int\_{-\infty}^\infty e^{itx}d\De\_n(x)=c\de\_n+c|t|\de\_n\sqrt{\ln\frac1{\de\_n}}. 
\end{equation\*}
Letting $t=1$ here and in (2), we see from the second line in (2) that 
$$\de\_n\sqrt{\ln\frac1{\de\_n}}\gg\frac{\ln\ln n}{\ln n},$$ 
whence the first inequality in (1) immediately follows. 


It appears that similar techniques should work for a somewhat wide class of distributions, like the ones referenced by the OP. 


