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 Ukraine 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), τ, such that W_τ has the same distribution as X,

[math]\displaystyle{ \operatorname{E}[\tau] = \operatorname{E}[X^2] }[/math]

and

[math]\displaystyle{ \operatorname{E}[\tau^2] \leq 4 \operatorname{E}[X^4]. }[/math]

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

[math]\displaystyle{ S_n = X_1 + \cdots + X_n. }[/math]

Then there is a sequence of stopping times τ₁ ≤ τ₂ ≤ ... such that the [math]\displaystyle{ W_{\tau_{n}} }[/math] have the same joint distributions as the partial sums S_n and τ₁, τ₂ − τ₁, τ₃ − τ₂, ... are independent and identically distributed random variables satisfying

[math]\displaystyle{ \operatorname{E}[\tau_n - \tau_{n - 1}] = \operatorname{E}[X_1^2] }[/math]

and

[math]\displaystyle{ \operatorname{E}[(\tau_{n} - \tau_{n - 1})^2] \leq 4 \operatorname{E}[X_1^4]. }[/math]

References

• Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc.. ISBN 0-471-00710-2.  (Theorems 37.6, 37.7)