Skorokhod embedding theorem

The Skorokhod embedding theorem represents a centered probability distribution $\mu$ on the real numbers with finite variance as $B_T\sim \mu$ for a stopping time $T$ of Brownian motion. One may choose $T$ so that the stopped process $(B_{t\wedge T})$ is uniformly integrable and $\mathbb{E}T=\int x^2\mu (dx)$.

If a random walk has independent identically distributed centered steps of variance ${\sigma }^2<\infty$, there are increasing Brownian stopping times $T_n$ such thatand the increments $T_n-T_{n-1}$ are independent and identically distributed with mean ${\sigma }^2$. Apply the one-step embedding repeatedly using the strong Markov property.

The law of large numbers gives $T_n/n\to {\sigma }^2$. Brownian maximal estimates then showin probability. Since $B_{n{\sigma }^2}/\sqrt{n}\sim N(0,{\sigma }^2)$, the embedded random walk satisfies the central limit theorem.