Editing Stochastic process (section)

===Measure theory and probability theory===
At the [[International Congress of Mathematicians]] in [[Paris]] in 1900, [[David Hilbert]] presented a list of [[Hilbert's problems|mathematical problems]], where his sixth problem asked for a mathematical treatment of physics and probability involving [[axiom]]s.<ref name="Bingham2000"/> Around the start of the 20th century, mathematicians developed measure theory, a branch of mathematics for studying integrals of mathematical functions, where two of the founders were French mathematicians, [[Henri Lebesgue]] and [[Émile Borel]]. In 1925, another French mathematician [[Paul Lévy (mathematician)|Paul Lévy]] published the first probability book that used ideas from measure theory.<ref name="Bingham2000"/>

In the 1920s, fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as [[Sergei Bernstein]], [[Aleksandr Khinchin]],{{efn|The name Khinchin is also written in (or transliterated into) English as Khintchine.<ref name="Doob1934">{{cite journal|last1=Doob|first1=Joseph|title=Stochastic Processes and Statistics|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=20|issue=6|year=1934|pages=376–379|doi=10.1073/pnas.20.6.376|pmid=16587907|pmc=1076423|bibcode=1934PNAS...20..376D|doi-access=free}}</ref>}} and [[Andrei Kolmogorov]].<ref name="Cramer1976"/> Kolmogorov published in 1929 his first attempt at presenting a mathematical foundation, based on measure theory, for probability theory.<ref name="KendallBatchelor1990page33">{{cite journal|last1=Kendall|first1=D. G.|last2=Batchelor|first2=G. K.|last3=Bingham|first3=N. H.|last4=Hayman|first4=W. K.|last5=Hyland|first5=J. M. E.|last6=Lorentz|first6=G. G.|last7=Moffatt|first7=H. K.|last8=Parry|first8=W.|last9=Razborov|first9=A. A.|last10=Robinson|first10=C. A.|last11=Whittle|first11=P.|title=Andrei Nikolaevich Kolmogorov (1903–1987)|journal=Bulletin of the London Mathematical Society|volume=22|issue=1|year=1990|page=33|issn=0024-6093|doi=10.1112/blms/22.1.31}}</ref> In the early 1930s, Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as [[Eugene Slutsky]] and [[Nikolai Smirnov (mathematician)|Nikolai Smirnov]],<ref name="Vere-Jones2006page1">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=1|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=978-0471667193}}</ref> and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.<ref name="Doob1934"/><ref name="Vere-Jones2006page4">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=4|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=978-0471667193}}</ref>{{efn|Doob, when citing Khinchin, uses the term 'chance variable', which used to be an alternative term for 'random variable'.<ref name="Snell2005">{{cite journal|last1=Snell|first1=J. Laurie|title=Obituary: Joseph Leonard Doob|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|page=251|issn=0021-9002|doi=10.1239/jap/1110381384|doi-access=free}}</ref> }}