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 Wτ has the same distribution as X,
![{\displaystyle \operatorname {E} [\tau ]=\operatorname {E} [X^{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42e52cf17b3d9136392b26ac7749ffe80dbca9b0)
and
![{\displaystyle \operatorname {E} [\tau ^{2}]\leq 4\operatorname {E} [X^{4}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e967b262e6d3cca547f7bb58c33a8515f6287c3d)
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
![{\displaystyle S_{n}=X_{1}+\cdots +X_{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a943da25d80015eff727ede1ee64f34086d1b20)
Then there is a sequence of stopping times τ1 ≤ τ2 ≤ ... such that the
have the same joint distributions as the partial sums Sn and τ1, τ2 − τ1, τ3 − τ2, ... are independent and identically distributed random variables satisfying
![{\displaystyle \operatorname {E} [\tau _{n}-\tau _{n-1}]=\operatorname {E} [X_{1}^{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21b862bed565a3341e5a3fd80c6e27ed808d236f)
and
![{\displaystyle \operatorname {E} [(\tau _{n}-\tau _{n-1})^{2}]\leq 4\operatorname {E} [X_{1}^{4}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/541eb1f10ed0f1fe0946b8b8c0e50f6ba3dd72b8)
References
- Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN 0-471-00710-2. (Theorems 37.6, 37.7)