5 The central limit theorem

Lemma 5.1

Let \(X\) be a real random variable with characteristic function \(\phi \) with \(\mathbb {E}[\vert X \vert ^n] {\lt} \infty \) and let \(k \le n\). Then the \(k\)^th derivative of \(\phi \) is \(\phi ^{(k)}(t) = \mathbb {E}[i X]^k e^{i t X}\)   .

Proof ▶

Lemma 5.2

Let \(X\) be a real random variable with characteristic function \(\phi \), with \(\mathbb {E}[\vert X \vert ^n] {\lt} \infty \). Then as \(t \to 0\),

\begin{align*} \phi (t) = \sum _{k=0}^n \frac{(it)^k \mathbb {E}[X^k]}{k!} + o(t^n) \: . \end{align*}

Proof ▶

Theorem 5.3 Central limit theorem

Let \(X_1, X_2, \ldots \) be i.i.d. real random variables with mean 0 and variance 1, and let \(Z\) be a random variable with law \(\mathcal N(0,1)\). Then

\begin{align*} \frac{1}{\sqrt{n}}\sum _{k=1}^n X_k \xrightarrow {d} Z \: . \end{align*}

Proof ▶

Let \(S_n = \frac{1}{\sqrt{n}}\sum _{k=1}^n X_k\). Let \(\phi \) be the characteristic function of \(X_k\). By Lemma 4.5 and 4.6, the characteristic function of \(S_n\) is \(\phi _n(t) = (\phi (n^{-1/2}t))^n\).

By Lemma 5.2,

\begin{align*} \phi _n(t) = (\phi (n^{-1/2}t))^n = \left(1 - \frac{1}{2n}t^2 + o(\frac{1}{n})\right)^n \to _{n \to +\infty } e^{-t^2/2} \: . \end{align*}

Since the r.h.s. is the characteristic function of \(Z\), we conclude that \(\frac{1}{\sqrt{n}}\sum _{k=1}^n X_k \xrightarrow {d} Z\) by Lemma 4.14.