= Solution
Write $S_n=n^{-1/2}\sum_{k=1}^nX_k$. By <independence of random variables>, its <moment-generating function> is
$$
M_{S_n}(t)=M_X(t/\sqrt n)^n.
$$
For each fixed real $t$, the expansion in part (i) yields
$$
\log M_{S_n}(t)
=n\left(\frac{t^2}{2n}+o(n^{-1})\right)
\longrightarrow\frac{t^2}{2}.
$$
For $t=0$ the equality is immediate. Exponentiation gives $M_{S_n}(t)\to e^{t^2/2}$, the <moment-generating function of a standard normal variable>. The functions are finite on a common neighborhood of zero, so the <convergence theorem for moment-generating functions> proves
$$
\boxed{S_n\xrightarrow dN(0,1).}
$$
This proves the <central limit theorem> under the exponential-moment hypothesis of this part.
Back to article page
