ben ebsworth
∫Mathematics

Random Walk

Stochastic walks converging to the Gaussian via the central limit theorem.

knobs
knobs

150 unbiased random walks. The cloud of paths spreads like √t — the hallmark of diffusive processes.

The Core Idea

A random walk is the simplest model of a process that accumulates uncertainty over time. At each step, you move by a random amount — up or down, left or right. After N steps, your position is the sum of N independent random variables. The central limit theorem guarantees that this sum, properly scaled, converges to a Gaussian distribution. But individual paths? Wildly unpredictable.

The Mathematics

A simple random walk is defined by:

Sₙ = X₁ + X₂ + ... + Xₙ

where each Xᵢ is drawn independently from some distribution (e.g., Xᵢ ∈ {-1, +1} with equal probability, or Xᵢ ~ N(0, σ²) for a Gaussian walk).

Key properties:

  • Expected position: E[Sₙ] = n·μ where μ is the drift (mean step size). With zero drift, the expected position is always the origin — even as individual walks wander far.
  • Variance: Var(Sₙ) = n·σ² — uncertainty grows linearly with time.
  • Standard deviation: σₙ = σ√n — the typical distance from the origin grows as the square root of time.

This √t scaling is the signature of diffusion: heat spreading through a rod, ink dispersing in water, stock prices fluctuating. It's why your GPS accuracy degrades as √(time) if you lose satellite lock.

knobs

Adding positive drift (μ = 0.3). The cloud still spreads, but the centre of mass drifts upward.

Try adjusting the drift slider — the expected path becomes a straight line with slope μ, while the cloud still spreads around it. This is exactly how financial models treat stock prices: drift is expected return, volatility is risk.

The arc sine law

Here's a paradox: in a symmetric random walk, the most likely scenario is that one side (positive or negative) dominates for almost the entire duration. The time spent above zero follows the arc sine distribution — it's U-shaped, meaning extreme splits (90/10, 95/5) are more probable than balanced ones (50/50). This is not intuitive, and it has profound implications for gambling, trading, and any sequential decision process.

knobs

Higher volatility — the spreading envelope widens, showing how σ scales the √t diffusion rate.

From walks to Brownian motion

Take the step size to zero while speeding up time proportionally, and the random walk converges to Brownian motion (the Wiener process W(t)) — a continuous, nowhere-differentiable path with independent Gaussian increments. Brownian motion is the building block of stochastic calculus (Itō's lemma, the Black-Scholes equation, the Fokker-Planck equation).

Applications

  • Finance: Stock prices are often modelled as geometric random walks (log-normal increments). The drift is the expected return; volatility is the risk.
  • Physics: Brownian motion of particles, diffusion equations, polymer chain conformations.
  • Biology: Bacterial chemotaxis (biased random walks toward nutrients).
  • Computer science: Randomised algorithms, network packet routing, Markov chain Monte Carlo.

Further reading

  • Random walk (Wikipedia)
  • Feller, An Introduction to Probability Theory and Its Applications, Vol. 1 — Chapter III on random walks is a masterpiece.
  • Mandelbrot, The Variation of Certain Speculative Prices (1963) — the classic challenge to Gaussian assumptions in finance.