Central Limit Theorem Variation for Chi Square distribution???

If this question fits into Homework Help, please move it over there. I'm not too sure.

I encountered the following problem:

Now, this problem seems fairly similar to a simple proof the central limit theorem. I am damn sure that this problem involves finding the mgf of D2, evaluating it and saying that it is the same as the mgf of a chi square function.

Can you help me out with setting up the equation? that'll be a big help! Thank you!

I've given it an attempt, but one attempt at setting up the mgf is all that i ask. Also, if there is any other way which does not involve mgf (i being wrong), please mentione that! Thank you!

Re: Central Limit Theorem Variation for Chi Square distribution???

I have seen a proof of this theorem, which has been proved by assuming the value of the parameter k = 2. Essentially, it just means that the theorem has been proved using ony the asumption tht only two events may occur.

p1 + p2 = 1 and n1 + n2 = n

Using this and substituting in the value of D2, we arrive at a uncture where n1 is defined as sum of j from 1 to n of Y_{ij}, where Y_{ij} = 1 is A_{1} occurs on the jth repetition and 0 elsewhere.

Now, central limit theorem is used over the variable n1, and if n is large, it has approximately a normal distribution.

I distinctly remember someone mentioning the use of Principle of Mathematical Induction to solve this problem. Can anyone of you solve this problem using the PMI? Will be a big help!!!! No mgf involved till now!!!