# Weak Convergence

Here is the problem (not homework),

Let $U_1,\cdots,U_n$ be i.i.d. uniform$(-n,n)$ random variables. For $-n<a<b<n$, we set $1_{U_i}(a,b)$ be the indicator function such that $1_{U_i}=1$ if $U_i\in(a,b)$ and 0 otherwise. What is approximate distribution as n large of $U_1+,\cdots,+U_n$.

I computed the characteristic function of $U_1+,\cdots,+U_n$, i.e., $\phi(t) = \left(\frac{sin(nt)}{nt}\right)^n$, but I don't know how to get the final result. By the way, I have no idea to use the hint in the problem. Please provide me some hints or references. Thanks!

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First, do the I read this correctly that you have $n$ instances of this random variable $U$, and also that $-n$ and $n$ are the bounds of the uniform distribution that the $U$s are instances of? –  Peter Ellis Sep 11 '12 at 10:14
Second, I don't see how the a, b, and the indicator function for $U$ being between them are relevant to the distribution of the sum of the $U$s. What is the reason for thinking they are relevant? –  Peter Ellis Sep 11 '12 at 10:16
Hint 1: Central limit theorem. Hint 2 (R): n=1000; sim = numeric(10000); for(i in 1:10000) sim[i]=sum(runif(n,-n,n)); hist(sim); hist(rnorm(10000,0,n^1.5/sqrt(3)),add=T) –  user10525 Sep 11 '12 at 10:27
@Procrastinator PeterEllis, thanks for your help. Since $Var(U_1 +\cdots+U_n) = 2n^3 \to \infty$ as $n \to \infty$, I am not pretty sure I can use the central limit theorem directly. –  Jim Sep 11 '12 at 11:52
@Jim There is no need for calculating the variance of the sum. You just need to know the variance of a single variable. The approximation is hidden in my second hint. This is a $\mbox{Normal}(\mu=0,\sigma=n^{3/2}/\sqrt{3})$. Try to check the CLT and verify the details. –  user10525 Sep 11 '12 at 11:58
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in general, what is the approximate distribution of the sum of $n$ independent and identically distributed random variables for large $n$?