Embed any zero-mean distribution as a Brownian motion stopped at a random time τ. Visualize the optional stopping theorem, martingale convergence, and the connection between random walks and arbitrary distributions.
Skorokhod embedding theorem (1965): Given any distribution μ with E[X]=0, E[X²]<∞, there exists a stopping time τ such that B_τ ~ μ and E[τ]=E[X²]=Var(X). The embedding can be constructed via Az´ema-Yor (1979): τ=inf{t: S_t ≥ Ψ(B_t)} where S_t=max_{s≤t}B_s. Consequences: Donsker's invariance principle (CLT via BM), concentration inequalities, functional form of OST. Martingale connection: stopped martingale M^τ is still a martingale.