The Central Limit Theorem
The √n law: averages don't just converge, they converge at a quantified rate to a universal shape. This subchapter states it, proves the binomial case with full computation, draws the convergence to scale, and marks precisely where the theorem's license ends.
In this subchapter
7.3.1Statement, and why √n
Let X₁, X₂, … be iid with mean m and finite variance σ² > 0. Then the normalized sums
converge in distribution to the standard normal: P(Zn ≤ t) → Φ(t) for every t. Equivalently (Portmanteau), E[g(Zn)] → E[g(Z)] for every bounded continuous g.
Why √n and not n? The strong law (7.2) says the deviation X̄n − m shrinks like o(1); the CLT refines: the typical size of the deviation is σ/√n — Chebyshev's bound said ≤ on average; the CLT says the distribution of the rescaled deviation settles into a fixed shape. The √n is forced by variance bookkeeping: Var(X̄n) = σ²/n (7.2 Move: the cross terms die by independence), so a scale-preserving normalization must multiply by exactly √n. The content of the theorem is not the scaling but the shape: why Gaussian, for every law with two moments, whatever its skew, bumps, or atoms — that is the remarkable part, and it is a local phenomenon: each variable contributes a vanishingly small share of the total, and only two moments' worth of its shape survives the aggregation. The proof formalizes exactly this "shape information is destroyed, moment information survives".
7.3.2The engine: characteristic functions
φX(t) = E[eitX] — the Fourier transform of the law: for a density, ∫ eitx f(x) dx; for atoms, the sum; always well-defined since |eitX| = 1 ≤ 1 (no integrability hypothesis needed — contrast the moment generating function, which needs E[etX] < ∞, a real constraint). Two facts make φ the tool: uniqueness (φ determines the law — the CDF-version of the π–λ uniqueness of 4.2 Theorem 3, proved the same way) and continuity (pointwise convergence of φ's to a continuous limit that is itself a φ identifies the convergence in distribution — Lévy's continuity theorem, stated here, proved in full generality in any graduate text; we flag it as the one external crane this chapter uses).
For independent X, Y: φX+Y(t) = E[eitXeitY] = φX(t)φY(t) — expectations factor by 6.1 Theorem 2. This is the entire reason characteristic functions run limit proofs: independence turns into multiplication, and multiplication turns into Taylor-exponential asymptotics. The mgf, where it exists, does the same — the price is its existence; the Cauchy mgf does not exist on any interval, its φ does (e−|t|, 5.5).
Center and scale once: Yi = (Xi − m)/σ, so E[Yi] = 0, Var = 1, and Zn = (Y₁ + ⋯ + Yn)/√n. Its characteristic function:
by the factoring lemma (n factors of an independent sum, scaled). Now the Taylor expansion at 0 — the ε-management center of the proof: since E[Y²] < ∞, φY(s) = 1 + isE[Y] − s²E[Y²]/2 + o(s²) = 1 − s²/2 + o(s²) as s → 0 (the o(s²) is itself a theorem — that second moments suffice for the two-term expansion. The standard proof truncates: on {|Y| ≤ K} the Taylor remainder is bounded by s³E[|Y|³; |Y|≤K] ≤ s³K·E[Y²], which is o(s²); on {|Y| > K} the remainder is bounded by a constant times s²E[Y²; |Y|>K], made small by choosing K large — an ε-management exercise in exactly the style of 5.2); the flag: we use the truncation pattern, not re-derive it). Setting s = t/√n and raising to the n-th power:
— the classical exponential limit (calculus: (1 + a/n + o(1/n))n → ea). The limit is continuous and is the standard normal's φ (a Gaussian integral you have computed since second year). Lévy's theorem converts: Zn converges in distribution to standard normal. ∎
The expansion kept two pieces of the law: E[Y] (killed by centering) and E[Y²] (normalized to 1). Everything else — skew, tails, atoms, bumps — lives in the o(s²) term and is annihilated by the (·)n with s = t/√n. This is the mathematical content of "the Gaussian is universal": the CLT is a forgetting theorem. It is also exactly why the hypothesis "finite variance" is load-bearing: the second moment is the entire surviving cargo; with E[X²] = ∞, the Taylor expansion collapses and the limit is not Gaussian (§5).
7.3.3De Moivre–Laplace, computed honestly
For Sn the number of heads in n fair flips: (Sn − n/2)/√(n/4) converges in distribution to standard normal.
Each flip: Xi = ±1/2 (centered), E[Xi²] = 1/4, and — by direct multiplication — φX(t) = cos(t/2) (the two-atom expectation: (eit/2 + e−it/2)/2). Then
using cos(s) = 1 − s²/2 + o(s²) — the Taylor expansion with the series you can verify by hand, no truncation lemmas needed because the cosine's expansion is absolute and exact. This is the honest computational skeleton of every CLT: for your own working problems, run this same computation with your own φ. ∎
7.3.4The figure, to scale
7.3.5Where the license ends
Iid with tail P(X > t) ~ t−α, 0 < α < 2 (5.5's Student/Pareto class): no finite variance, the CLT fails — and fails with a known successor: the normalized sums (Sn − nm)/n1/α converge to an α-stable law (for 1 < α < 2; for α ≤ 1 no centering term exists). The stable laws are the fixed points of the aggregation operation — self-similar under addition, with Gaussian = the α = 2 endpoint. Practical translation: aggregating heavy-tailed risks does not Gaussianize them; it rescales them into another heavy-tailed shape, and confidence intervals built on √n logic are not conservative — they are wrong, by factors that grow with n. The Berry–Esseen bound below makes the failure quantitative in the finite-variance case; beyond it, stable-limit theory is the correct language (flagged, not developed).
There is a universal constant C (best known bounds: 0.4 < C < 0.56) with
— the CLT's error, uniform in t, controlled by the third moment. Read it as a license with a meter: the normal approximation is good when n is large relative to the skewness cubed; for sums of skewed payoffs (option books, insurance) the meter reads expensive and the third-moment correction matters. The bound's shape explains a quant's folklore ("CLT converges slowly for skew") as mathematics: the error is exactly skew-per-√n.
CLT convergence is pointwise in t, and in the tails it is arbitrarily bad in relative terms: at t = 4, Φ says ~3×10−5; the normal approximation to a sum of bounded variables at 4σ has relative error governed by large-deviation theory (not the CLT — the CLT licenses the center of the distribution, t = O(1)). VaR at 99.9% computed by "normal approximation" is a CLT applied where its theorem says least; the error is not a rounding issue but a different-limit issue (large deviations, flagged). The tool for the tail is either exact computation, the stable/large-deviation asymptotics, or simulation — with the convergence-mode discipline of 7.1 deciding what "simulation validates" can even mean.
7.3.6The asymptotic toolbox, assembled
Chapter 7 is the practitioner's core, and it is complete: LLN (what averages do — converge a.s., under tails), modes of convergence (what convergence can even mean — four ways, no reversals), CLT (the rate and shape — Gaussian, √n, second moments), plus the two failure regimes (heavy tails → stable laws; tails-in-t → large deviations) that mark the boundary of the license. Every statistical method, risk formula, and backtest methodology in daily use is an application of one of these theorems under one of these hypotheses — and the course's claim, now substantiated, is that knowing which theorem and which hypothesis is the difference between modeling and folklore.
The Gaussian is what remains of a distribution after aggregation forgets everything but its first two moments. When the aggregation cannot forget (heavy tails), the Gaussian never arrives.
One structure remains, and it is the one practitioners use under the name "updating": conditioning. Chapter 8 builds it honestly — conditional expectation as a σ-algebra projection (not a formula), the Radon–Nikodym theorem that makes it exist, martingales as the dynamics of fair games, and the closure: where to take the theory next.