Central limit result for a transformation of iid $U(0,1)$ samples

Question

Let $X_1,\cdots,X_n$ be iid $U(0,1)$ samples, how can I establish a central limit result result for $$Y_n=\frac{(X_1+\cdots+X_n)^2}{n(X_1^2+\cdots+X_n^2)}?$$ or how can I express $Y_n$ as sum of some independent random variables?

Answer 1

If you are trying to see whether $Y_n$ converges (in some sense) then, using the law of large numbers may be handy here. Notice that $$ Y_n =\frac{\Big(\frac1n\sum^n_{k=1}X_k\Big)^2}{\frac1n\sum^n_{k=1}X^2_k} $$ The numerator converges (a.s) to $(E[X_1])^2=1/4$ and the denominator converges to $E[X^2_1]=1/3$ (a.s); hence $Y_n\xrightarrow{n\rightarrow\infty}3/4$ a.s.

Te get something related to the normal distribution, you may try the so-called delta method. Set $Z_n=X_n-\frac12$ so that $\{Z_n\}$ is still i.i.d with $E[Z_n]=0$ and $\operatorname{var}(Z_n)=\frac{1}{12}$. Denote $\overline{Z_n}=\frac12\sum^n_{k=1}Z_n$ and $\overline{Z^2_n}=\frac1n\sum^n_{k=1}Z^2_n$. Then $$ Y_n = g(\overline{Z_n},\overline{Z^2_n}) $$ where $g(u,v)=\frac{(u+\frac12)^2}{v+u+\frac14}$. Notice that The Delta--method implies that $$ \sqrt{n}\Big(Y_n -g(0,\frac{1}{12})\big)\Longrightarrow g'(0,\frac{1}{12})N((0,0),\Sigma) $$ where $$ \Sigma =\begin{pmatrix} \frac{1}{12} & 0\\ 0 & \frac{1}{80}-\frac{1}{12^2} \end{pmatrix} $$ The rest is just a little bit of calculus and arithmetic manipulations.