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.

korokhod'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",

:$mathbb\{E\}\; [\; au]\; =\; mathbb\{E\}\; [X^\{2\}]$

and

:$mathbb\{E\}\; [\; au^\{2\}]\; leq\; 4\; mathbb\{E\}\; [X^\{4\}]\; .$

(Naturally, the above inequality is trivial unless "X" has finite fourth moment.)

korokhod's second embedding theorem

Let "X"₁, "X"₂, ... 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 non-decreasing (a.k.a. weakly increasing) sequence "τ"₁, "τ"₂, ... of stopping times such that the $W\_\{\; au\_\{n$ have the same joint distributions as the partial sums "S"_"n" and "τ"₁, "τ"₂ − "τ"₁, "τ"₃ − "τ"₂, ... are independent and identically distributed random variables satisfying

:$mathbb\{E\}\; [\; au\_\{n\}\; -\; au\_\{n\; -\; 1\}]\; =\; mathbb\{E\}\; [X\_\{1\}^\{2\}]$

and

:$mathbb\{E\}\; [(\; au\_\{n\}\; -\; au\_\{n\; -\; 1\})^\{2\}]\; leq\; 4\; mathbb\{E\}\; [X\_\{1\}^\{4\}]\; .$

References

* (Theorems 37.6, 37.7)