Skorokhod's embedding theorem

Remarks

For a law on two points $a < 0 < b$, the exit time of $B$ from $(a, b)$ works. The optional stopping theorem for the martingales $B_t$ and $B_t^2 - t$ gives the right weights and $E[T] = -ab$. The general case writes $\mu$ as a mixture of such two-point laws, or builds $T$ by a sequence of exit times from nested intervals. Applied repeatedly, the theorem gives stopping times $0 = T_0 \le T_1 \le T_2 \le \cdots$ with i.i.d. increments such that $(B_{T_n})_{n \ge 0}$ has the law of a random walk with step law $\mu$. This embedding is one route to Donsker's theorem and to limit laws for random walks. The mean zero condition is necessary: if $E[T] < \infty$, then Wald's identities give $E[B_T] = 0$ and $E[B_T^2] = E[T]$.

Going deeper

For a student

The theorem lets one transfer questions about sums of independent variables to questions about a single Brownian path. A random walk becomes Brownian motion sampled at random times $T_n$, and the strong law of large numbers gives $T_n / n \to \int x^2 \, d\mu$. So the walk and the path stay close on the scale $\sqrt{n}$, which yields functional limit theorems. Many explicit constructions of $T$ exist, with different extra properties.