Skorokhod's embedding theorem

In mathematics and probability theory, Skorokhod's embedding theorem is either or both of two theorems that allow one to regard any suitable collection of random variables as a Wiener process (Brownian motion) evaluated at a collection of stopping times. Both results are named for the Ukrainian mathematician A.V. Skorokhod.

Skorokhod's first embedding theorem
Let X be a real-valued random variable with expected value 0 and finite variance; let W denote a canonical real-valued Wiener process. Then there is a stopping time (with respect to the natural filtration of W), &tau;, such that W&tau; has the same distribution as X,


 * $$\mathbb{E}[\tau] = \mathbb{E}[X^{2}]$$

and


 * $$\mathbb{E}[\tau^{2}] \leq 4 \mathbb{E}[X^{4}].$$

(Naturally, the above inequality is trivial unless X has finite fourth moment.)

Skorokhod's second embedding theorem
Let X1, X2, ... be a sequence of independent and identically distributed random variables, each with expected value 0 and finite variance, and let


 * $$S_{n} = X_{1} + \cdots + X_{n}.$$

Then there is a non-decreasing (a.k.a. weakly increasing) sequence &tau;1, &tau;2, ... of stopping times such that the $$W_{\tau_{n}}$$ have the same joint distributions as the partial sums Sn and &tau;1, &tau;2 &minus; &tau;1, &tau;3 &minus; &tau;2, ... are independent and identically distributed random variables satisfying


 * $$\mathbb{E}[\tau_{n} - \tau_{n - 1}] = \mathbb{E}[X_{1}^{2}]$$

and


 * $$\mathbb{E}[(\tau_{n} - \tau_{n - 1})^{2}] \leq 4 \mathbb{E}[X_{1}^{4}].$$