Misplaced Pages

Skorokhod's embedding theorem

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

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 Wτ has the same distribution as X,

E ⁡ [ τ ] = E ⁡ [ X 2 ] {\displaystyle \operatorname {E} =\operatorname {E} }

and

E ⁡ [ τ 2 ] ≤ 4 E ⁡ [ X 4 ] . {\displaystyle \operatorname {E} \leq 4\operatorname {E} .}

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 . {\displaystyle S_{n}=X_{1}+\cdots +X_{n}.}

Then there is a sequence of stopping times τ1 ≤ τ2 ≤ ... such that the W τ n {\displaystyle W_{\tau _{n}}} have the same joint distributions as the partial sums Sn and τ1, τ2 − τ1, τ3 − τ2, ... are independent and identically distributed random variables satisfying

E ⁡ [ τ n − τ n − 1 ] = E ⁡ [ X 1 2 ] {\displaystyle \operatorname {E} =\operatorname {E} }

and

E ⁡ [ ( τ n − τ n − 1 ) 2 ] ≤ 4 E ⁡ [ X 1 4 ] . {\displaystyle \operatorname {E} \leq 4\operatorname {E} .}

References

  • Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN 0-471-00710-2. (Theorems 37.6, 37.7)
Categories: