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.