Random Walks — Section 13: Stochastic Processes

A simple random walk takes a step of size $\pm 1$ at each tick, with equal probability:

S_n = X_1 + X_2 + \cdots + X_n, \quad X_i \in \{-1, +1\}

The expected position is $E[S_n] = 0$ and the variance grows linearly: $\mathrm{Var}(S_n) = n$ . So typical excursions scale like $\sqrt{n}$ .

Random walks model anything that drifts due to accumulated random shocks — gambler's bankroll, asset prices in the simplest model, particle motion. Their continuous-time scaling limit is Brownian motion: take steps of size $\sqrt{\Delta t}$ over intervals of length $\Delta t$ and let $\Delta t \to 0$ .

A subtlety: simple random walks on $\mathbb{Z}$ are recurrent (return to origin infinitely often) in 1D and 2D, but transient in 3D. "Drunk man on a line eventually finds his keys; drunk bird in 3-space is lost forever."

A simple random walk takes a step of size $\pm 1$ at each tick, with equal probability:

S_n = X_1 + X_2 + \cdots + X_n, \quad X_i \in \{-1, +1\}

The expected position is $E[S_n] = 0$ and the variance grows linearly: $\mathrm{Var}(S_n) = n$ . So typical excursions scale like $\sqrt{n}$ .