Skorokhod embedding of a centered random walk
= Skorokhod embedding of a centered random walk
{c}
If a random walk has independent identically distributed centered steps of variance $\sigma^2<\infty$, there are increasing Brownian stopping times $T_n$ such that
$$
(B_{T_n})_{n\geq0}\overset d=(S_n)_{n\geq0},
$$
and 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.