Editing Stochastic process (section)

====Separability====
'''Separability''' is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a [[separable space]],{{efn|The term "separable" appears twice here with two different meanings, where the first meaning is from probability and the second from topology and analysis. For a stochastic process to be separable (in a probabilistic sense), its index set must be a separable space (in a topological or analytic sense), in addition to other conditions.<ref name="Skorokhod2005page93"/>}} which means that the index set has a dense countable subset.<ref name="Adler2010page14"/><ref name="Ito2006page32">{{cite book|author=Kiyosi Itō|title=Essentials of Stochastic Processes|url=https://books.google.com/books?id=pY5_DkvI-CcC&pg=PR4|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3898-3|pages=32–33}}</ref>

More precisely, a real-valued continuous-time stochastic process <math>X</math> with a probability space <math>(\Omega,{\cal F},P)</math> is separable if its index set <math>T</math> has a dense countable subset <math>U\subset T</math> and there is a set <math>\Omega_0 \subset \Omega</math> of probability zero, so <math>P(\Omega_0)=0</math>, such that for every open set <math>G\subset T</math> and every closed set <math>F\subset \textstyle R =(-\infty,\infty) </math>, the two events <math>\{ X_t \in F \text{ for all }  t \in G\cap U\}</math> and <math>\{ X_t \in F \text{ for all }  t \in G\}</math> differ from each other at most on a subset of <math>\Omega_0</math>.<ref name="GikhmanSkorokhod1969page150">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=150}}</ref><ref name="Todorovic2012page19">{{cite book|author=Petar Todorovic|title=An Introduction to Stochastic Processes and Their Applications|url=https://books.google.com/books?id=XpjqBwAAQBAJ&pg=PP5|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-9742-7|pages=19–20}}</ref><ref name="Molchanov2005page340">{{cite book|author=Ilya Molchanov|title=Theory of Random Sets|url=https://books.google.com/books?id=kWEwk1UL42AC|year=2005|publisher=Springer Science & Business Media|isbn=978-1-85233-892-3|page=340}}</ref>
The definition of separability{{efn|The definition of separability for a continuous-time real-valued stochastic process can be stated in other ways.<ref name="Billingsley2008page526">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|pages=526–527}}</ref><ref name="Borovkov2013page535">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=535}}</ref>}} can also be stated for other index sets and state spaces,<ref name="GusakKukush2010page22">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 22</ref> such as in the case of random fields, where the index set as well as the state space can be <math>n</math>-dimensional Euclidean space.<ref name="AdlerTaylor2009page7"/><ref name="Adler2010page14"/>

The concept of separability of a stochastic process was introduced by [[Joseph Doob]],.<ref name="Ito2006page32"/> The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process.<ref name="Billingsley2008page526"/> Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable.<ref name="Doob1990page56">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=56}}</ref> A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification.<ref name="Ito2006page32"/><ref name="Todorovic2012page19"/><ref name="Khoshnevisan2006page155">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|page=155}}</ref> Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.<ref name="Skorokhod2005page93"/>