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 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 variance; let W denote a canonical real-valued Wiener process. Then there is a stopping time, τ, such that Wτ has the same distribution as X,
and

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
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
and