---
tag: XHTYZ7ER4Z
kind: statement
role: theorem
name: Skorokhod's embedding theorem
msc: [60J65, 60G40, 60G50]
uses: [22735FGJV6, WND87FBAKC, 059TJN0NZ1, DC9HZT2HVJ, 04VZE3FZXF, 818832PMSV]
about: [22735FGJV6, WND87FBAKC]
sources:
  - url: https://en.wikipedia.org/wiki/Skorokhod%27s_embedding_theorem
same_as:
  wikidata: Q7536350
  wikipedia: Skorokhod's embedding theorem
---

## Statement

Let $(B_t)_{t \ge 0}$ be a standard one-dimensional [[Brownian motion|22735FGJV6]] with $B_0 = 0$, and let $(\mathcal{F}_t)$ be the filtration generated by $B$.
Let $\mu$ be a [[probability measure|059TJN0NZ1]] on $\R$ with $\int x^2 \, d\mu(x) < \infty$ and $\int x \, d\mu(x) = 0$.
Then there is a [[stopping time|WND87FBAKC]] $T$ for $(\mathcal{F}_t)$ such that $B_T$ has law $\mu$ and
$$E[T] = \int_{\R} x^2 \, d\mu(x).$$

## Remarks

For a law on two points $a < 0 < b$, the exit time of $B$ from $(a, b)$ works.
The [[optional stopping theorem|DC9HZT2HVJ]] 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|04VZE3FZXF]] with step law $\mu$.
This embedding is one route to [[Donsker's theorem|818832PMSV]] 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 $E[B_T^2] = E[T]$.

## Explanations

### L1 human

Any centred distribution with finite variance can be produced by watching a Brownian path and stopping it at a cleverly chosen moment.
The decision to stop uses only the path seen so far.
For a fair coin with values $\pm 1$, one stops the first time the path reaches $1$ or $-1$.

### L2 human

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.