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·μwhere 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.
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.
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.