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I'm working on the question below and I appreciate if you can guide me on how I can solve it.

Here is the question:

Consider $X_j$'s j = 1, 2, ... as independent bernulli random variables. These random variables, although are independent, but are not identically distributed(so then I thought that I should use Lindeberg-Feller CLT). We knoe $P(X_j = 1) = P_j$ and we know that $P_j$'s are bounded between (0,1) {open interval}. I'm trying to show $(S_n - ES_n)/\sqrt{VarS_n}$ converges to N(0,1) in distribution.

Here is what I've done so far:

1) Consider $Y_j = (X_j - P_j)/(P_j*(1 - P_j))$ ==> EY_j = 0 {first condition of lindeberg-feller thm satisfied}

2) $\sum_{j = 11}^{n} E(Y_j^2) = \sum Var(Y_j^2)$ and is finite

3) I need to check the lindeberg-feller condition: $lim \sum_{j = 1}^{n} E(Y_j^2.1_{|Y_j| \gt \epsilon}) = 0$ ??? I don't know how to prove the third one.

Could you please guide me whether I'm on the right track to solve this question and also how I can show that lindeberg condition is satisfied.

Thanks a lot for your help.

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up vote 1 down vote accepted

You'll have trouble if e.g. $P_j \to 0$ very rapidly. In fact if $\sum_{j=2}^\infty P_j < 1$, there is a nonzero probability that all $S_n = S_1$, and there is no hope of getting a normal limit. Or did you mean that there is some $\delta > 0$ such that all $P_j \in [\delta, 1-\delta]$?

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