Skorokhod's embedding theorem - Alchetron, the free social encyclopedia

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,

E [ τ ] = E [ X 2 ]

and

E [ τ 2 ] ≤ 4 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 .

Then there is a sequence of stopping times τ₁ ≤ τ₂ ≤ ... such that the W τ 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

E [ τ n − τ n − 1 ] = E [ X 1 2 ]

and

E [ ( τ n − τ n − 1 ) 2 ] ≤ 4 E [ X 1 4 ] .

References

Skorokhod's embedding theorem Wikipedia

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Contents

Skorokhod's first embedding theorem

Skorokhod's second embedding theorem

References