Two random variables generated with common random varibales I have two random variables (RVs) (in general I have more $U_1,\cdots,U_N$) which are defined as
$$U_1=\sum_{i=1,i\neq 1}^{N}x_i$$
$$U_2=\sum_{i=1,i\neq 2}^{N}x_i$$
where $x_i$s are i.i.d. RVs. Since $U_1$ and $U_2$ have common $x_i$s, $U_1$ and $U_2$ may be correlated. 
If $N$ is sufficiently large, I can get distributions for $U_1$ and $U_2$ by using the central limit theorem (CLT). In this case, is it reasonable to consider (or can be shown ???) that $U_1$ and $U_2$ are uncorrelated? 
 A: I definitely would not expect $U_1$ and $U_2$ to be uncorrelated: they should be identical in the natural scaling limit if the $X_i$'s are reasonably nice. Just look at the difference: $U_1 - U_2 = X_2 - X_1$, which is small compared to the sum $U_1$ or $U_2$, and totally disappears in the centered/scaled sums: 
\begin{equation}
\frac{U_1 - U_2}{\sqrt{N}} = \frac{X_2 - X_1}{\sqrt{N}} \to 0.
\end{equation}
Said another way: $U_1$ and $U_2$ are identical to the total sum $U = \sum_{i=1}^N X_i$ in the limit with the CLT scaling. 
One can actually compute the correlation directly. The variance of both $U_1$ and $U_2$ is $(N-1) \sigma^2$, (writing $\sigma^2 = Var(X)$). Thus 
$Corr(U_1, U_2) = \frac{1}{(N-1)^2 \sigma^4} Cov(U_1, U_2) = \frac{(N-2)\sigma^2}{(N-1) \sigma^2} \to 1$
as $N \to \infty$.
I'm not sure if you care about the case where $X$ doesn't satisfy the hypotheses of a CLT, but I suppose something interesting could happen in this case. If $\mathbb{E}|X| < \infty$, then the natural scaling for $U$ is $\frac{1}{N} U$, and again $U_1$ and $U_2$ will be essentially identical. But maybe something strange could happen if $\mathbb{E}X = \infty$...
