---
tag: H018Y2JG4Q
kind: proof
name: Proof of the central limit theorem by characteristic functions
msc: [60F05]
proves: 0EPZC4Z1FC
uses: [0AJN7WHBJZ, 0XCE070V8B, 0MYDNRT7PP, 02JGYG0T7M, 0NS0TDA4E2, 0FPWF7TTMB, 0T8BYAFEXX, 0JDAQHWA9N, 07RXBVTVZB, 007QED3E0Y, BYNV84PF86, 0NHGY0N0JB, 0H5ZAZCX2T, 0WM7DJCM6G, 0P7PTC8DDE]
sources:
  - url: https://en.wikipedia.org/wiki/Central_limit_theorem
    locator: Proof of classical CLT
  - url: https://sites.math.duke.edu/~rtd/PTE/PTE5_011119.pdf
    locator: Theorem 3.4.1 and Lemma 3.4.3
---

## Proof

We show that the characteristic function of the normalized sum tends to $e^{-t^2/2}$. Then we use Lévy's continuity theorem.

1. Let $Y_i = (X_i - \mu)/\sigma$. For a Borel set $B$, $\set{Y_i \in B} = \set{X_i \in g^{-1}(B)}$ with the continuous map $g(x) = (x - \mu)/\sigma$. So the $Y_i$ are [[independent|0XCE070V8B]] and identically distributed, like the [[i.i.d.|0AJN7WHBJZ]] variables $X_i$.
2. By [[linearity of expectation|0P7PTC8DDE]] and the definition of the [[variance|007QED3E0Y]], $E[Y_1] = 0$ and $E[Y_1^2] = \sigma^2/\sigma^2 = 1$.
3. Let $Z_n = (X_1 + \dots + X_n - n\mu)/(\sigma\sqrt{n}) = (Y_1 + \dots + Y_n)/\sqrt{n}$. Note that $Z_n = \sqrt{n}(\bar{X}_n - \mu)/\sigma$.
4. Let $\varphi$ be the [[characteristic function|0MYDNRT7PP]] of $Y_1$. It is also the characteristic function of each $Y_i$, since it depends only on the law.
5. By the definition, $\varphi_{Z_n}(t) = E[e^{i(t/\sqrt{n})(Y_1 + \dots + Y_n)}]$. By the [[characteristic function of a sum of independent random variables|02JGYG0T7M]], $\varphi_{Z_n}(t) = \varphi(t/\sqrt{n})^n$.
6. Claim: $\abs{e^{ix} - 1 - ix} \le x^2/2$ for real $x$. By the [[fundamental theorem of calculus|0NHGY0N0JB]], on real and imaginary parts, $e^{ix} - 1 - ix = \int_0^x i(e^{iu} - 1) \, du$. Also $\abs{e^{iu} - 1} = \abs{\int_0^u i e^{iv} \, dv} \le \abs{u}$.
7. So the modulus in step 6 is at most $\abs{\int_0^x \abs{u} \, du} = x^2/2$, which proves the claim.
8. Claim: $\abs{e^{ix} - 1 - ix + x^2/2} \le \min(\abs{x}^3/6, \, x^2)$ for real $x$. Again $e^{ix} - 1 - ix + x^2/2 = \int_0^x i(e^{iu} - 1 - iu) \, du$, so by step 7 the modulus is at most $\abs{\int_0^x u^2/2 \, du} = \abs{x}^3/6$.
9. Also, by step 7, the modulus is at most $\abs{e^{ix} - 1 - ix} + x^2/2 \le x^2$. This proves the claim.
10. For real $s$, let $q(s) = E\left[\min\left(\abs{s} \abs{Y_1}^3/6, \, Y_1^2\right)\right]$. By step 2, $\varphi(s) - 1 + s^2/2 = E[e^{isY_1} - 1 - isY_1 + s^2 Y_1^2/2]$.
11. By step 8 with $x = sY_1$, $\abs{\varphi(s) - 1 + s^2/2} \le E\left[\min(\abs{s}^3 \abs{Y_1}^3/6, \, s^2 Y_1^2)\right] = s^2 q(s)$.
12. Claim: $q(s) \to 0$ as $s \to 0$. Let $s_j \to 0$. The integrand tends to $0$ at each $\omega$, since it is at most $\abs{s_j} \abs{Y_1}^3/6$. It is bounded by $Y_1^2$, which is integrable.
13. By [[dominated convergence|BYNV84PF86]], $q(s_j) \to 0$. Since the sequence $s_j$ was arbitrary, the claim holds.
14. Claim: $\abs{e^{-s^2/2} - 1 + s^2/2} \le s^4/8$. By [[Taylor's theorem with the Lagrange remainder|0H5ZAZCX2T]], for $y = s^2/2 > 0$ there is $\xi \in (0, y)$ with $e^{-y} = 1 - y + e^{-\xi} y^2/2$. Since $0 < e^{-\xi} < 1$, the error is at most $y^2/2 = s^4/8$.
15. Steps 11 and 14 give $\abs{\varphi(s) - e^{-s^2/2}} \le s^2 q(s) + s^4/8$.
16. Claim: for complex $z, w$ with $\abs{z} \le 1$ and $\abs{w} \le 1$, $\abs{z^n - w^n} \le n \abs{z - w}$. Indeed $z^n - w^n = (z - w) \sum_{k=0}^{n-1} z^k w^{n-1-k}$, and each of the $n$ terms has modulus at most $1$.
17. Fix $t \in \R$. By the [[product of exponentials|0WM7DJCM6G]], $e^{-t^2/2} = (e^{-t^2/(2n)})^n$. Also $\abs{\varphi(t/\sqrt{n})} \le 1$ and $0 < e^{-t^2/(2n)} \le 1$.
18. By steps 5, 16, 17 and 15 with $s = t/\sqrt{n}$, $\abs{\varphi_{Z_n}(t) - e^{-t^2/2}} \le n \abs{\varphi(t/\sqrt{n}) - e^{-t^2/(2n)}} \le t^2 q(t/\sqrt{n}) + t^4/(8n)$.
19. By step 13, the right side tends to $0$. So $\varphi_{Z_n}(t) \to \psi(t) = e^{-t^2/2}$ for each $t \in \R$.
20. The function $\psi$ is continuous at $0$. By [[Lévy's continuity theorem|0FPWF7TTMB]], $\psi$ is the characteristic function of a random variable $W$, and $Z_n \to W$ [[in distribution|0JDAQHWA9N]].
21. By the [[characteristic function of the normal distribution|0NS0TDA4E2]], $\psi$ is also the characteristic function of $\mathcal{N}(0, 1)$. Since [[the characteristic function determines the distribution|0T8BYAFEXX]], $W \sim \mathcal{N}(0, 1)$, and its cumulative distribution function is $\Phi$.
22. The [[standard normal density|07RXBVTVZB]] is bounded by $1/\sqrt{2\pi}$. So $\abs{\Phi(x) - \Phi(y)} \le \abs{x - y}/\sqrt{2\pi}$, and $\Phi$ is continuous at every point.
23. By steps 20 to 22, $P(Z_n \le x) \to \Phi(x)$ for each $x \in \R$. This is the claim of the theorem.

## Idea

After centering and scaling, $Y_1$ has characteristic function $1 - s^2/2 + o(s^2)$, the same to second order as $e^{-s^2/2}$. Raising to the power $n$ at $s = t/\sqrt{n}$ multiplies the error by $n$, but the error is $o(1/n)$, so the limit is $e^{-t^2/2}$. Lévy's continuity theorem turns this into convergence of the distribution functions.