Skorokhod's embedding theorem

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In mathematics and probability theory, Skorokhod's embedding theorem refers to either or both of two theorems that allows 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.

[edit] 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,

$\mathbb{E}[\tau] = \mathbb{E}[X^{2}]$

and

$\mathbb{E}[\tau^{2}] \leq 4 \mathbb{E}[X^{4}].$

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

[edit] 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

$S_{n} = X_{1} + \dots + X_{n}.$

Then there is a non-decreasing (a.k.a. weakly increasing) sequence τ₁, τ₂, ... of stopping times such that the $W_{\tau_{n}}$ 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