$\{X_i\}$ i.d normally distributed r.v.$E(X_n)=0,Var(X_n)=1+1/n$,judge whether $\{X_n\}$ satisfies CLT?

Question

Let $\{X_i\}$ be independent normally distributed random variables.$E(X_n)=0,Var(X_n)=1+1/n$,judge whether $\{X_n\}$ satisfies CLT(central limit theorem) conclusion,i.e,$S_n/\sqrt {Var(S_n)}\to\mathcal N(0,1)$ in distribution?

Attempts:I tried the Lindeberg condition,but I think it is hard to estimate this sum:$\frac{1}{c_n^2}\sum_{k=1}^n\int_{|X_k|\geq\epsilon c_n} x_k^2dF_k(x)$,which $\geq\epsilon^2\sum_{k=1}^nP(|X_k|\geq\epsilon c_n)$,but that seems unable to give an answer.

Also I tried the Liapounov's condition,but I haven't found the proper $\delta$ yet.

Answer 1

There's no need to use Lindeberg condition or Liapounov's condition in this particular situation. From the properties of normal distribution, we have:

$S_n=X_1+\ldots+X_n\sim\mathcal{N}(E(S_n),Var(S_n)^2)$

$E(S_n)=E(X_1)+\ldots+E(X_n)=0$

$Var(S_n)^2=Var(X_1)^2+\ldots+var(X_n)^2=(1+1)^2+\ldots+(1+\frac{1}{n})^2$

Then we just get $S_n/\sqrt{Var(S_n)}\sim\mathcal{N}(0,1)$ directly.

Answer 2

Hint: Show that the moment-generating function of $X_n$ converges to a normal one.

Could also show that the CDF converges, but up to you. Lindeberg would be overkill.