Editing Stochastic process (section)

====Markov processes====
Markov processes and Markov chains are named after [[Andrey Markov]] who studied Markov chains in the early 20th century. Markov was interested in studying an extension of independent random sequences. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a [[weak law of large numbers]] without the independence assumption,<ref name="GrinsteadSnell1997page464">{{cite book|author1=Charles Miller Grinstead|author2=James Laurie Snell|title=Introduction to Probability|url=https://archive.org/details/flooved3489|year=1997|publisher=American Mathematical Soc.|isbn=978-0-8218-0749-1|pages=[https://archive.org/details/flooved3489/page/n473 464]–466}}</ref><ref name="Bremaud2013pageIX">{{cite book|author=Pierre Bremaud|title=Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues|url=https://books.google.com/books?id=jrPVBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3124-8|page=ix}}</ref><ref name="Hayes2013">{{cite journal|last1=Hayes|first1=Brian|title=First links in the Markov chain|journal=American Scientist|volume=101|issue=2|year=2013|pages=92–96|doi=10.1511/2013.101.92}}</ref> which had been commonly regarded as a requirement for such mathematical laws to hold.<ref name="Hayes2013"/> Markov later used Markov chains to study the distribution of vowels in [[Eugene Onegin]], written by [[Alexander Pushkin]], and proved a [[central limit theorem]] for such chains.

In 1912, Poincaré studied Markov chains on [[finite group]]s with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by [[Paul Ehrenfest|Paul]] and [[Tatyana Ehrenfest]] in 1907, and a branching process, introduced by [[Francis Galton]] and [[Henry William Watson]] in 1873, preceding the work of Markov.<ref name="GrinsteadSnell1997page464"/><ref name="Bremaud2013pageIX"/> After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by [[Irénée-Jules Bienaymé]].<ref name="Seneta1998">{{cite journal|last1=Seneta|first1=E.|title=I.J. Bienaymé [1796-1878]: Criticality, Inequality, and Internationalization|journal=International Statistical Review / Revue Internationale de Statistique|volume=66|issue=3|year=1998|pages=291–292|issn=0306-7734|doi=10.2307/1403518|jstor=1403518}}</ref> Starting in 1928, [[Maurice Fréchet]] became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.<ref name="GrinsteadSnell1997page464"/><ref name="BruHertz2001">{{cite book|last1=Bru|first1=B.|title=Statisticians of the Centuries|last2=Hertz|first2=S.|chapter=Maurice Fréchet|year=2001|pages=331–334|doi=10.1007/978-1-4613-0179-0_71|isbn=978-0-387-95283-3}}</ref>

[[Andrei Kolmogorov]] developed in a 1931 paper a large part of the early theory of continuous-time Markov processes.<ref name="Cramer1976"/><ref name="KendallBatchelor1990page33"/> Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as [[Norbert Wiener]]'s work on Einstein's model of Brownian movement.<ref name="KendallBatchelor1990page33"/><ref name="BarbutLocker2016page5">{{cite book|author1=Marc Barbut|author2=Bernard Locker|author3=Laurent Mazliak|title=Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters|url=https://books.google.com/books?id=lSz_vQAACAAJ|date= 2016|publisher=Springer London|isbn=978-1-4471-7262-8|page=5}}</ref> He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes.<ref name="KendallBatchelor1990page33"/><ref name="Skorokhod2005page146">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|page=146}}</ref> Independent of Kolmogorov's work, [[Sydney Chapman (mathematician)|Sydney Chapman]] derived in a 1928 paper an equation, now called the [[Chapman–Kolmogorov equation]], in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement.<ref name="Bernstein2005">{{cite journal|last1=Bernstein|first1=Jeremy|title=Bachelier|journal=American Journal of Physics|volume=73|issue=5|year=2005|pages=398–396|issn=0002-9505|doi=10.1119/1.1848117|bibcode=2005AmJPh..73..395B}}</ref> The differential equations are now called the Kolmogorov equations<ref name="Anderson2012pageVII">{{cite book|author=William J. Anderson|title=Continuous-Time Markov Chains: An Applications-Oriented Approach|url=https://books.google.com/books?id=YpHfBwAAQBAJ&pg=PR8|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3038-0|page=vii}}</ref> or the Kolmogorov–Chapman equations.<ref name="KendallBatchelor1990page57">{{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=57|issn=0024-6093|doi=10.1112/blms/22.1.31}}</ref> Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in the 1930s, and then later Eugene Dynkin, starting in the 1950s.<ref name="Cramer1976"/>