Proof of the Central Limit Theorem

We prove the Central Limit Theorem using characteristic functions (Fourier transforms), the standard approach in modern probability theory.

Proof

Theorem (CLT): Let $X_1, X_2, \ldots$ be i.i.d. with $E[X_i] = 0$ , $\operatorname{Var}(X_i) = \sigma^2 > 0$ . Then $S_n = \frac{X_1 + \cdots + X_n}{\sigma\sqrt{n}} \xrightarrow{d} N(0,1)$ .

Step 1: Characteristic functions.

The characteristic function of a random variable $X$ is $\varphi_X(t) = E[e^{itX}]$ . By Levy's continuity theorem, $X_n \xrightarrow{d} X$ if and only if $\varphi_{X_n}(t) \to \varphi_X(t)$ pointwise for all $t \in \mathbb{R}$ .

The characteristic function of $N(0,1)$ is $\varphi_{Z}(t) = e^{-t^2/2}$ .

Step 2: Compute the characteristic function of $S_n$ .

Let $Y_i = X_i / (\sigma\sqrt{n})$ , so $S_n = Y_1 + \cdots + Y_n$ . By independence: $\varphi_{S_n}(t) = \prod_{i=1}^n \varphi_{Y_i}(t) = [\varphi_{Y_1}(t)]^n = \left[\varphi_{X_1}\left(\frac{t}{\sigma\sqrt{n}}\right)\right]^n$

Step 3: Taylor expansion of the characteristic function.

Since $E[X_1] = 0$ and $E[X_1^2] = \sigma^2$ , the characteristic function of $X_1$ has the expansion: $\varphi_{X_1}(s) = E[e^{isX_1}] = 1 + is \cdot E[X_1] + \frac{(is)^2}{2} E[X_1^2] + o(s^2)$ $= 1 - \frac{\sigma^2 s^2}{2} + o(s^2) \quad \text{as } s \to 0$

Substituting $s = t/(\sigma\sqrt{n})$ : $\varphi_{X_1}\left(\frac{t}{\sigma\sqrt{n}}\right) = 1 - \frac{\sigma^2}{2} \cdot \frac{t^2}{\sigma^2 n} + o\left(\frac{1}{n}\right) = 1 - \frac{t^2}{2n} + o\left(\frac{1}{n}\right)$

Step 4: Take the $n$ -th power.

$\varphi_{S_n}(t) = \left[1 - \frac{t^2}{2n} + o\left(\frac{1}{n}\right)\right]^n$

Using the fundamental limit $\lim_{n\to\infty}(1 + a_n/n)^n = e^a$ when $a_n \to a$ :

$\lim_{n \to \infty} \varphi_{S_n}(t) = e^{-t^2/2}$

More rigorously: write $\varphi_{X_1}(t/\sigma\sqrt{n}) = 1 + w_n$ where $w_n = -t^2/(2n) + o(1/n)$ . Then $\log(1 + w_n) = w_n + O(w_n^2) = -t^2/(2n) + o(1/n)$ , so $n \log(1 + w_n) \to -t^2/2$ , giving $\varphi_{S_n}(t) = e^{n\log(1+w_n)} \to e^{-t^2/2}$ .

Step 5: Apply Levy's continuity theorem.

Since $\varphi_{S_n}(t) \to e^{-t^2/2} = \varphi_{Z}(t)$ pointwise and $e^{-t^2/2}$ is continuous at $t = 0$ , Levy's continuity theorem guarantees $S_n \xrightarrow{d} Z \sim N(0,1)$ . $\square$

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RemarkWhy characteristic functions?

The proof via characteristic functions is elegant because: (1) the characteristic function of a sum of independent variables is the product of their characteristic functions, converting convolution to multiplication; (2) pointwise convergence of characteristic functions is equivalent to convergence in distribution (Levy's theorem); (3) the Taylor expansion naturally produces the Gaussian $e^{-t^2/2}$ .