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.

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'',

:$\operatorname$tau= \operatorname ^2/math>

and

:\operatorname tau^2\leq 4 \operatorname ^4

Skorokhod'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 sequence of stopping times ''τ''₁ ≤ ''τ''₂ ≤ ... such that the $W_$ 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

:\operatorname tau_n - \tau_= \operatorname _1^2/math>

and

:\operatorname \tau_ - \tau_)^2\leq 4 \operatorname _1^4

References

* {{cite book , last=Billingsley , first=Patrick , title=Probability and Measure , publisher=John Wiley & Sons, Inc. , location=New York , year=1995 , isbn=0-471-00710-2 (Theorems 37.6, 37.7)

Probability theorems
Wiener process
Ukrainian inventions