Discrete random walks

Independent step effects add in the mean, while their uncertainty adds in variance.

Let Sₙ be the sum of independent steps X₁ through Xₙ, where each step is +1 with probability p and −1 otherwise.

E[Sₙ]=n(2p−1); for a fair walk the mean is 0 and Var(Sₙ)=n.

Typical distance grows like √n, not n, because positive and negative steps cancel partly.

Write each step as +1 or −1 and add them. For a fair walk, positive and negative means cancel, but squared uncertainty cannot cancel, so n independent step variances add to n.

Standard deviation is the square root of variance, which makes typical displacement grow like sqrt(n). After 100 fair steps, being around 10 places from the start is ordinary, while 100 places would require every step to point the same way.

Positions share history: S_20 contains all ten steps already inside S_10. That is why increments from disjoint time intervals can be independent even though the positions S_10 and S_20 are not.

Random walks model cumulative measurement error, inventory changes, and simplified price moves. They are useful baselines because the variance scaling is explicit; persistent trends, boundaries, and dependent steps must be added when reality shows them.

Brownian motion as a scaling limit

Brownian motion has continuous paths but new Gaussian noise at every time scale, making paths almost surely nowhere differentiable.

Standard Brownian motion has W₀=0, independent increments, and Wₜ−Wₛ~N(0,t−s) for t>s.

Its covariance is Cov(Wₛ,Wₜ)=min(s,t), and increments scale like √Δt.

Brownian motion is an idealized model; real data can have jumps, memory, bounded speeds, or changing variance.

Brownian motion keeps the same square-root scaling while making time continuous. Over a gap Δt, the increment is normal with variance Δt and standard deviation sqrt(Δt), no matter where the interval starts.

Independent increments mean future changes are independent of the completed past, not that the future position forgets the current position. Wₜ equals its current level plus a fresh increment.

Continuity and roughness coexist: the path has no jumps, yet it wiggles so intensely at every scale that an ordinary velocity derivative almost surely does not exist. Stochastic calculus was built for exactly this behavior.

Diffusion models add carefully scaled Gaussian noise over many small steps until data becomes nearly noise, then learn a reverse denoising direction. Brownian scaling explains why each noise increment uses sqrt(Δt), while the learned drift tells how the distribution should move.

Practical deep dive for this lesson

Scale a random walk into Brownian units

To obtain a nontrivial continuous limit, speed up the number of steps while shrinking each step by the square root of that number.

After n fair steps, variance is n. Dividing position by sqrt(n) makes variance one, so the scaled process neither explodes nor collapses as n grows.

For time t, use roughly nt steps and scale by 1/sqrt(n). The resulting variance is nt/n≈t, matching the defining Brownian increment variance.

This scaling explains the sqrt(dt) noise in simulations. Using dt would make accumulated variance vanish; using an unscaled fixed jump would make paths explode as time steps become numerous.