# Central limit theorem basic application

i do have a question regarding the following problem:

Given random Variables $Y_k$ iid with $E(Y_k) = 0$ and $E(Y_k^2)=1$ and $Z_k$ with $P(Z_k = k)=P(Z_k = -k)$ Set $X_k = Y_k + Z_k$ and $S_n = X_1+X_2+...+X_n$ Show that $\sqrt n S_n$ converges in distribution to a standard normal distribution.

To me, this clearly looks like an application of the CLT, but i cant figure out a way to do it. The solution would follow from the CLT if i calculate the expected value and the variance, but it dont know how to do that (yet).

Thanks for helping me

• assuming $Y_i$ and $Z_j$ are independent, this is straightforward. Calculate $E[X_k]$ using linearity of expectation. Calculate variance using $\mbox{Var}(X_k)=\mbox{Var}(Y_k)+\mbox{Var}(Z_k)$. – Alex R. Jan 12 '17 at 22:24