Skip to content
Stochastic Calculus / Brownian Motion: Construction and Properties

Random Walk Limits and the Multivariate Normal Distribution

Log-returns of equity prices are approximately normally distributed over short horizons, and the sum of nn independent daily log-returns converges to a normal as nn grows. This is the Central Limit Theorem, and it is the mathematical reason Black-Scholes models log-prices with Brownian motion. But the CLT proves convergence of one-dimensional distributions. To model a path of prices — a function of time, not just a single future value — we need to know the joint distribution of infinitely many increments. That joint structure is captured by the multivariate normal distribution.

This note establishes the CLT (via the characteristic-function proof that Lawler calls the cleanest route), contrasts it with the Poisson limit that arises in jump models (Merton's jump-diffusion), and then develops the multivariate normal distribution in the matrix-algebra language needed to construct Brownian motion in the next lesson and correlated multi-asset models in Lesson 8.

Learning objectives. By the end of this note you should be able to:
  • State the CLT, sketch its proof via characteristic functions, and identify the condition (finite variance) on which it rests.
  • Explain why the Poisson limit arises instead of a normal when increments are rare and large, and identify when a jump-diffusion model falls into the Poisson regime.
  • Define a multivariate normal distribution via linear combinations of independent standard normals and identify its covariance matrix.
  • Prove that any covariance matrix is symmetric and positive semi-definite, and state the converse.
  • Derive the characteristic function ϕ(θ)=e−12θΓθ⊤\phi(\theta) = e^{-\frac{1}{2}\theta \Gamma \theta^\top} of a mean-zero multivariate normal with covariance Γ\Gamma.
  • State Donsker's theorem (invariance principle) as the path-level extension of the CLT and identify what it says about scaled random walks.

In this lesson

  1. The Central Limit Theorem
  2. The Poisson Limit
  3. The Multivariate Normal Distribution
  4. Donsker's Invariance Principle (Sketch)

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or explore the curriculum.