Skorokhod's embedding theorem

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 Wτ has the same distribution as X, {displaystyle operatorname {E} [tau ]=operatorname {E} [X^{2}]} and {displaystyle operatorname {E} [tau ^{2}]leq 4operatorname {E} [X^{4}].} 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 {displaystyle S_{n}=X_{1}+cdots +X_{n}.} 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 Sn and τ1, τ2 − τ1, τ3 − τ2, ... are independent and identically distributed random variables satisfying {displaystyle operatorname {E} [tau _{n}-tau _{n-1}]=operatorname {E} [X_{1}^{2}]} and {displaystyle operatorname {E} [(tau _{n}-tau _{n-1})^{2}]leq 4operatorname {E} [X_{1}^{4}].} References Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN 0-471-00710-2. (Theorems 37.6, 37.7) Categories: Probability theoremsWiener processUkrainian inventions