1. ## Limiting Distribution

Let X_1,X_2,...,X_n be i.i.d. random variables from N(1,1) distribution. Find the limiting distribution of the ran. var.

W_n = sqrt(n)*[(X_1 + X_2 + ... + X_n - n)] / [((X_1 - 1)^2) + ... + ((X_n - 1)^2)].

We have recently seen this limiting distributions but haven't solved enough problems about it and i don't know which theorem i should apply for a problem(though i guess central limit theorem should work for this problem). Please help me with this. Thanks!

2. Start by finding the moment generating function of Wn, ie $E(e^{W_n t})$, once you have found it, take it's limit as n approaches infinity.

3. It looks like your trying to get a T, but there are 3 problems

$\sum_{i=1}^nX_i\sim N(n,n)$

thus ${\sum_{i=1}^nX_i-n\over\sqrt{n}}\sim N(0,1)$

and $\sum_{i=1}^n(X_i-1)^2\sim \chi_n^2$

NOW a t with a degrees of freedom is a ${N(0,1)\over\sqrt{\chi^2_a/a}}\sim t_a$

So, you're missing another sqrt of n, the sqrt on the denominator
BUT moreover, these two sums don't seem to be independent.
Maybe we can use that $\bar X$ is independent of $S^2$
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Got it, I was trying to find it's EXACT distribution, but limiting is easy.
Use two facts, which I already said...

${\sum_{i=1}^nX_i-n\over\sqrt{n}}\sim N(0,1)$

$\sum_{i=1}^n(X_i-1)^2\sim\chi_n^2$

Thus by the SLLN ${\sum_{i=1}^n(X_i-1)^2\over n}\to E(\chi_1^2)=1$

take the ratio and you are done, the limiting result is a N(0,1)

${{\sum_{i=1}^nX_i-n\over\sqrt{n}}\over {\sum_{i=1}^n(X_i-1)^2\over n}}\to {N(0,1)\over 1}$

This is the CLT plus Slutsky.
Actually the numerator is a N(0,1), so it's not really a CLT.