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.

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 (with respect to the natural filtration of W), τ, such that W_τ has the same distribution as X,

and

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 ${\displaystyle W_{\tau _{n}}}$ 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

...
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