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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,

E ⁡ [ τ ] = E ⁡ [ X 2 ] {\displaystyle \operatorname {E} [\tau ]=\operatorname {E} [X^{2}]}

and

E ⁡ [ τ 2 ] ≤ 4 E ⁡ [ X 4 ] . {\displaystyle \operatorname {E} [\tau ^{2}]\leq 4\operatorname {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 + ⋯ + X n . {\displaystyle S_{n}=X_{1}+\cdots +X_{n}.}

Then there is a sequence of stopping times τ1 ≤ τ2 ≤ ... such that the W τ n {\displaystyle W_{\tau _{n}}} have the same joint distributions as the partial sums Sn and τ1, τ2 − τ1, τ3 − τ2, ... are independent and identically distributed random variables satisfying

E ⁡ [ τ n − τ n − 1 ] = E ⁡ [ X 1 2 ] {\displaystyle \operatorname {E} [\tau _{n}-\tau _{n-1}]=\operatorname {E} [X_{1}^{2}]}

and

E ⁡ [ ( τ n − τ n − 1 ) 2 ] ≤ 4 E ⁡ [ X 1 4 ] . {\displaystyle \operatorname {E} [(\tau _{n}-\tau _{n-1})^{2}]\leq 4\operatorname {E} [X_{1}^{4}].}

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