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,


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

and


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

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 sequence of stopping times &tau;1 &le; &tau;2 &le; ... 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


 * $$\operatorname{E}[\tau_n - \tau_{n - 1}] = \operatorname{E}[X_1^2]$$

and


 * $$\operatorname{E}[(\tau_{n} - \tau_{n - 1})^2] \le 4 \operatorname{E}[X_1^4].$$