Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 101 / 1 / a / Solution

For the central limit theorem, let $X_1,X_2,\dots$ be independent and identically distributed random variables with expectation $\mu$ and finite positive variance ${\sigma }^2$. ThenThus the centered, normalized sum has convergence in distribution to the standard normal distribution. The positive-variance assumption is needed for this normalization; if ${\sigma }^2=0$, each summand equals $\mu$ almost surely and the centered sum is identically zero.

Here is a characteristic function proof using only the finite second moment. Put $U=(X_1-\mu )/\sigma$, so $\mathbb{E}U=0$ and $\mathbb{E}{U}^2=1$. The elementary integral remainder identity gives $|e^{iz}-1-iz|\le z^2/2$ for real $z$. ConsequentlyBy the dominated convergence theorem, the characteristic function satisfies ${\phi }_U(t)=1-t^2/2+o(t^2)$ as $t\to 0$. For a fixed real $t$, independence therefore givesThe allowed complex-number limit makes this tend to $e^{-t^2/2}$, the characteristic function of the standard normal distribution. It is continuous at zero, so the Lévy continuity theorem proves the asserted convergence in distribution. No third moment or boundedness of the summands was used.