Start by finding the moment generating function of Wn, ie , once you have found it, take it's limit as n approaches infinity.
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!
It looks like your trying to get a T, but there are 3 problems
thus
and
NOW a t with a degrees of freedom is 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 is independent of
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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...
Thus by the SLLN
take the ratio and you are done, the limiting result is a N(0,1)
This is the CLT plus Slutsky.
Actually the numerator is a N(0,1), so it's not really a CLT.