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}].

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 τ₁ ≤ τ₂ ≤ ... 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)

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