Editing Stochastic process (section)

===Terminology===
The definition of a stochastic process varies,<ref name="FristedtGray2013page580">{{cite book|author1=Bert E. Fristedt|author2=Lawrence F. Gray|title=A Modern Approach to Probability Theory|url=https://books.google.com/books?id=9xT3BwAAQBAJ&pg=PA716|year= 2013|publisher=Springer Science & Business Media|isbn=978-1-4899-2837-5|page=580}}</ref> but a stochastic process is traditionally defined as a collection of random variables indexed by some set.<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/> The terms ''random process'' and ''stochastic process'' are considered synonyms and are used interchangeably, without the index set being precisely specified.<ref name="Kallenberg2002page24"/><ref name="ChaumontYor2012"/><ref name="AdlerTaylor2009page7"/><ref name="Stirzaker2005page45">{{cite book|author=David Stirzaker|title=Stochastic Processes and Models|url=https://books.google.com/books?id=0avUelS7e7cC|year=2005|publisher=Oxford University Press|isbn=978-0-19-856814-8|page=45}}</ref><ref name="Rosenblatt1962page91">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press|page=[https://archive.org/details/randomprocesses00rose_0/page/91 91]}}</ref><ref name="Gubner2006page383">{{cite book|author=John A. Gubner|title=Probability and Random Processes for Electrical and Computer Engineers|url=https://books.google.com/books?id=pa20eZJe4LIC|year=2006|publisher=Cambridge University Press|isbn=978-1-139-45717-0|page=383}}</ref> Both "collection",<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/> or "family" are used<ref name="Parzen1999"/><ref name="Ito2006page13">{{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|page=13}}</ref> while instead of "index set", sometimes the terms "parameter set"<ref name="Lamperti1977page1"/> or "parameter space"<ref name="AdlerTaylor2009page7"/> are used.

The term ''random function'' is also used to refer to a stochastic or random process,<ref name="GikhmanSkorokhod1969page1"/><ref name="Loeve1978">{{cite book|author=M. Loève|title=Probability Theory II|url=https://books.google.com/books?id=1y229yBbULIC|year=1978|publisher=Springer Science & Business Media|isbn=978-0-387-90262-3|page=163}}</ref><ref name="Brémaud2014page133">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-09590-5|page=133}}</ref> though sometimes it is only used when the stochastic process takes real values.<ref name="Lamperti1977page1"/><ref name="Ito2006page13"/> This term is also used when the index sets are mathematical spaces other than the real line,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 1</ref> while the terms ''stochastic process'' and ''random process'' are usually used when the index set is interpreted as time,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1"/><ref name="Bass2011page1">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=1}}</ref> and other terms are used such as ''random field'' when the index set is <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math> or a [[manifold]].<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="AdlerTaylor2009page7"/>