From Random Walk to Brownian Motion
Central Question
How can a jagged discrete path approach a continuous random process?
Why This Matters
A scaling limit connects discrete randomness to a continuous object while revealing that convergence of individual observations is not enough to control paths.
Learning Prompt
Motivate Brownian motion as the scaling limit of a symmetric random walk. Make me determine how space and time must be rescaled to keep the variance meaningful, and explain why interpolation is needed when comparing paths. Build the roles of Gaussian increments, independence, stationary increments and continuity. Distinguish convergence at fixed times from convergence of random paths, giving a conceptual account of what a functional limit theorem adds. Explore why continuity need not imply differentiability, and keep separate the construction of Brownian motion, its defining properties and the limiting argument.