Skorokhod's embedding theorem: Difference between revisions

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In [[mathematics]] and [[probability theory]], '''Skorokhod's embedding theorem''' refers tois either or both of two [[theorem]]s that allowsallow one to regard any suitable collection of [[random variable]]s as a [[Wiener process]] ([[Brownian motion]]) evaluated at a collection of [[stopping time]]s. Both results are named for the [[Ukraine|Ukrainian]] [[mathematician]] [[Anatoliy Skorokhod|A. V. Skorokhod]].

==Skorokhod's first embedding theorem==

Let ''X'' be a [[real number|real]]-valued random variable with [[expected value]] 0 and [[Wikt:finite|finite]] [[variance]]; let ''W'' denote a canonical real-valued Wiener process. Then there is a stopping time (with respect to the natural [[filtration (abstract algebra)|filtration]] of ''W''), ''τ'', such that ''W''_''τ'' has the same distribution as ''X'',

:<math>\mathbboperatorname{E}[\tau] = \mathbboperatorname{E}[X^{2}]</math>

and

:<math>\mathbboperatorname{E}[\tau^{2}] \leq 4 \mathbboperatorname{E}[X^{4}].</math>

(Naturally, the above inequality is trivial unless ''X'' has finite fourth [[moment (mathematics)|moment]].)

==Skorokhod's second embedding theorem==

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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

:<math>S_{n}S_n = X_{1}X_1 + \dotscdots + X_{n}X_n.</math>

Then there is a non-[[decreasing]]sequence (a.k.a.of weaklystopping increasing) sequencetimes ''&tau;''₁, &le; ''&tau;''₂, &le; ... of stopping times such that the <math>W_{\tau_{n}}</math> have the same joint distributions as the partial sums ''S''_''n'' and ''&tau;''₁, ''&tau;''₂ &minus; ''&tau;''₁, ''&tau;''₃ &minus; ''&tau;''₂, ... are independent and identically distributed random variables satisfying

:<math>\mathbboperatorname{E}[\tau_{n}tau_n - \tau_{n - 1}] = \mathbboperatorname{E}[X_{1}X_1^{2}]</math>

and

:<math>\mathbboperatorname{E}[(\tau_{n} - \tau_{n - 1})^{2}] \leqle 4 \mathbboperatorname{E}[X_{1}X_1^{4}].</math>

==References==

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

[[Category:Theorems in probability theory]]

[[Category:Wiener process]]

[[Category:Ukrainian inventions]]