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), τ, such that W_τ 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

$S_{n}=X_{1}+\dots +X_{n}.$

Then there is a non-decreasing (a.k.a. weakly increasing) sequence τ₁, τ₂, ... of stopping times such that the $W_{\tau _{n}}$ 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

$\mathbb {E} [\tau _{n}-\tau _{n-1}]=\mathbb {E} [X_{1}^{2}]$

and

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

References

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