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{"datasets_id": 161584, "wiki_id": "Q916989", "sp": 54, "sc": 68, "ep": 58, "ec": 133} | 161,584 | Q916989 | 54 | 68 | 58 | 133 | Socialist Alliance (Australia) | Indigenous rights & Anti-racism and immigrants rights | for indigenous Australians, particularly around the inquiries into the deaths-in-custody of TJ Hickey in Redfern and Mulrunji Doomadgee on Palm Island. In the case of Mulrunji, leading indigenous activist, academic and Socialist Alliance member Sam Watson played a key role in organising the protests that led to the re-opening of the inquiry.
Socialist Alliance also opposes the Federal Government's Northern Territory intervention, and helped to organise the 12 February 2008 protests outside Parliament House in Canberra. Anti-racism and immigrants rights Socialist Alliance has been able to build growing support among some ethnic community sectors in urban Australia such as among Somali |
{"datasets_id": 161584, "wiki_id": "Q916989", "sp": 58, "sc": 133, "ep": 62, "ec": 212} | 161,584 | Q916989 | 58 | 133 | 62 | 212 | Socialist Alliance (Australia) | Anti-racism and immigrants rights & Public services | youth, the Tamil community and from within the Latin American community. In the latter case, the Socialist Alliance has been an active supporter of the Bolivarian Revolution in Venezuela and is affiliated to the Australia Venezuela Solidarity Network.
Socialist Alliance members have also been involved in the struggle for refugee rights, opposing mandatory detention of illegal immigrants, and calling for Australia to pursue a more humane policy on refugees. Public services Socialist Alliance advocate the provision of quality public services by all levels of government, calling for increased funding in public education, healthcare, housing and transport. They also advocate expanding |
{"datasets_id": 161584, "wiki_id": "Q916989", "sp": 62, "sc": 212, "ep": 66, "ec": 144} | 161,584 | Q916989 | 62 | 212 | 66 | 144 | Socialist Alliance (Australia) | Public services & Social justice | the public sector with the nationalisation of large multinational corporations. Furthermore, the party calls for capitalist enterprises that have received taxpayer-subsidies to either repay their subsidies back to the taxpayers in full or be nationalised without compensation.
Socialist Alliance is involved in campaigns against privatisation like those planned by the New South Wales Government (for example electricity and prisons), alongside the Greens, unions, ALP members and community groups. They maintain that all privatisations must be reversed with nationalisation. Social justice Socialist Alliance is also active in a number of other social justice campaigns, including LGBTI rights, women's liberation, welfare rights, and |
{"datasets_id": 161584, "wiki_id": "Q916989", "sp": 66, "sc": 144, "ep": 70, "ec": 266} | 161,584 | Q916989 | 66 | 144 | 70 | 266 | Socialist Alliance (Australia) | Social justice & International solidarity | prison reform, as well as around local issues. After an editorial by OUTinPerth accusing socialists of taking over the movement for equal marriage rights, prominent LQBTIQ campaigner and Socialist Alliance member Farida Iqbal issued a reply arguing that the Socialist Alliance and others had played a prominent role in the Australian movement for marriage equality since it began in 2004. International solidarity The party also places a large emphasis on international socialist solidarity. It is actively involved in supporting many left-wing movements around the world, such as those relating to Venezuela and the Bolivarian Revolution in Latin America, Palestinian resistance, |
{"datasets_id": 161584, "wiki_id": "Q916989", "sp": 70, "sc": 266, "ep": 74, "ec": 500} | 161,584 | Q916989 | 70 | 266 | 74 | 500 | Socialist Alliance (Australia) | International solidarity & Criticism | Kurdish self-determination in North Syria. Socialist Alliance also actively campaigns in solidarity with international pro-democracy movements as far ranging as Latin America,
the Middle East,
Western Sahara,
Zimbabwe,
South East Asia, and elsewhere. Criticism Other political organisations on the Australian far left have criticise the Socialist Alliance project. Socialist Alternative, for example, contest that a sustained mass radicalisation had been born out of the anti-capitalist movement or that a significant layer of disillusioned ALP voters are willing to join a socialist electoral program. Socialist Alternative also criticises Socialist Alliance for what it perceives to be an over-emphasis on electoral work.
Upon its resignation from the |
{"datasets_id": 161584, "wiki_id": "Q916989", "sp": 74, "sc": 500, "ep": 74, "ec": 1202} | 161,584 | Q916989 | 74 | 500 | 74 | 1,202 | Socialist Alliance (Australia) | Criticism | Alliance, the former International Socialist Organisation accused the Democratic Socialist Perspective of what it deemed "disastrous decisions" such as declaring the Alliance a multi-tendency socialist party and adopting Green Left Weekly as the official paper, which the ISO saw as alienating other Alliance members and affiliates.
The Revolutionary Socialist Party (who formed in 2008 as a split from the DSP over debates about Socialist Alliance) accused the Alliance project of remaining "heavily dependent on the DSP’s political and organising efforts and fundraising." The RSP also (incorrectly) claimed that only the DSP remained an affiliate of the Alliance by 2008.
Due to the |
{"datasets_id": 161584, "wiki_id": "Q916989", "sp": 74, "sc": 1202, "ep": 74, "ec": 1395} | 161,584 | Q916989 | 74 | 1,202 | 74 | 1,395 | Socialist Alliance (Australia) | Criticism | similarity of their names and initials, Socialist Alliance and Socialist Alternative are frequently confused in the media and amongst critics of the social movements that they are involved in. |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 2, "sc": 0, "ep": 8, "ec": 292} | 161,585 | Q7566320 | 2 | 0 | 8 | 292 | South Berkeley, Berkeley, California | Demographics | South Berkeley, Berkeley, California South Berkeley is a neighborhood in the city of Berkeley, California. It extends roughly from Dwight Way to the city’s border with Oakland, between Telegraph Avenue in the east and either Sacramento Street or San Pablo Avenue in the west. It lies at an elevation of 102 feet (31 m). Demographics This neighborhood is the center for Berkeley's African-American community, with a population of 9,341 that is roughly 52% African American. Traditionally, it was considered to be the most economically depressed portion of Berkeley; however, as rent has risen in the city over the past several |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 8, "sc": 292, "ep": 12, "ec": 366} | 161,585 | Q7566320 | 8 | 292 | 12 | 366 | South Berkeley, Berkeley, California | Demographics & History | years, South Berkeley has become more attractive to students and other young people, and rents in the area have become comparable to other, more affluent Berkeley neighborhoods. South Berkeley is crisscrossed by AC Transit bus lines. History South Berkeley is part of the old Rancho San Antonio, approximately 45,000 acres of land granted by Don Pablo Vicente de Sola, Governor of Alta California, in 1820, to Luis Maria Peralta in recognition of his forty years of military service and his work in establishing the missions of Santa Clara, Santa Cruz, and San Jose. In 1842, Peralta divided his land between |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 12, "sc": 366, "ep": 12, "ec": 1004} | 161,585 | Q7566320 | 12 | 366 | 12 | 1,004 | South Berkeley, Berkeley, California | History | his four sons. José Domingo Peralta received the title to the northernmost portion, including present-day Berkeley and Albany.
By 1872, the Berkeley L.T.I. Association had mapped out the gridded streets of what is now South Berkeley; however, there were no houses yet.
In 1873, a house was built for Mark Ashby, and 10 years later, to the south, a house for his brother William Ashby. The house, in 1938, was advertised as a Friendly farm, where "FRIENDLY FARM - Applications now being accepted for spring semester, to start March 1st. Groups limited to 12 pre-school children Curriculum Includes painting, nature study, |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 12, "sc": 1004, "ep": 16, "ec": 226} | 161,585 | Q7566320 | 12 | 1,004 | 16 | 226 | South Berkeley, Berkeley, California | History & Places | music, habit training. Nominal rates. 2915 Deakin. Berkeley." ( The name of the street having been changed from North street )
During World War II, Camp Ashby, a camp for African American soldiers, was established in the area.
South Berkeley has been the East Bay mecca for sports, from competitive softball leagues, little league baseball, to the Midnight & Twilight Basketball League at Grove Park. Places Services and businesses located in South Berkeley include the Ashby BART station, the Shotgun Players theatre, La Peña Cultural Center, and the Berkeley Tool Lending Library. The Berkeley Bowl supermarket, which has one of the |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 16, "sc": 226, "ep": 20, "ec": 186} | 161,585 | Q7566320 | 16 | 226 | 20 | 186 | South Berkeley, Berkeley, California | Places & Historical Plaques | most extensive selections of produce and specialty foods in the Bay Area, operates one of its two stores there. The Ashby BART station hosts a flea market each weekend in its parking lot. The area is also home to Wat Mongkolratanaram, a Thai Buddhist temple that serves a Sunday brunch.
A 100-foot-long mural of South Berkeley history on Ashby Avenue at Ellis Street was painted in 2018 under the direction of muralist Edythe Boone. Historical Plaques The Berkeley Historical Plaque Project has, so far, commemorated six locations in South Berkeley:
At 2960 Sacramento Street, a plaque and nearby sculpture commemorate Dr. William |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 20, "sc": 186, "ep": 20, "ec": 934} | 161,585 | Q7566320 | 20 | 186 | 20 | 934 | South Berkeley, Berkeley, California | Historical Plaques | Byron Rumford's Pharmacy.
At 2643 Dana Street, a plaque memorializes the home and life of Anthony Boucher, Editor and Writer.
At 1500 Derby Street, a City of Berkeley Landmark plaque commemorates Longfellow School, an architecturally and culturally important public institution which currently houses the Longfellow Magnet Middle School.
At 2237 Carleton Street, a City of Berkeley Landmark plaque commemorates the Woodworth House, home and workspace for entomologist, naturalist, physicist, and inventor Charles W. Woodworth. Woodworth was a professor of entomology, and assisted in developing the University’s College of Agriculture and the City of Berkeley’s first public library.
At 3332 Adeline Street, a City |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 20, "sc": 934, "ep": 24, "ec": 144} | 161,585 | Q7566320 | 20 | 934 | 24 | 144 | South Berkeley, Berkeley, California | Historical Plaques & Parks and recreation | of Berkeley Landmark plaque commemorates the Lorin Theater, South Berkeley's first neighborhood theater, that, at one point, could seat up to 1,500 people.
At 3290 Adeline Street, a City of Berkeley Landmark plaque commemorates the South Berkeley Bank. In the early 20th century, this was one of two banks anchoring the busy Lorin business district’s streetcar intersection. Designed by John Galen Howard, the bank building shares an architect with multiple other buildings in Berkeley. Parks and recreation South Berkeley includes San Pablo Park (13 acres), Grove Park (3 acres), and several mini-parks. Grove Park is home to a nationally ranked men's |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 24, "sc": 144, "ep": 24, "ec": 745} | 161,585 | Q7566320 | 24 | 144 | 24 | 745 | South Berkeley, Berkeley, California | Parks and recreation | basketball team, the Berkeley All-Stars, coached by Bay Area basketball legend Eugene Evans. Grove Park and San Pablo Park have been the training grounds for many well-known athletes, including Don Barksdale, Claudell Washington, Phil Chenier, Shooty Babitt, Je'Rod Cherry and Jason Kidd. The tennis court area at Grove Park is now called William C. Charles Courts, named after the late "Mr. Charles" (also known as "the waving man"). He would stand in front of his house on the corner of Grove and Oregon Streets every morning from about 7 am to 10 am waving to passersby while saying "Keep smiling" |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 24, "sc": 745, "ep": 28, "ec": 365} | 161,585 | Q7566320 | 24 | 745 | 28 | 365 | South Berkeley, Berkeley, California | Parks and recreation & Notable residents | and "Have a beautiful day". He did this every day for 25 years. South Berkeley residents respected and appreciated Mr. Charles, who died in 1998, and the Grove Street Park tennis courts were named in his honor. Notable residents William Bryon Rumford - (February 2, 1908 – June 12, 1986) - First Black person elected to a state public office in Northern California - William was the first African American elected to a state public office in Northern California. He became the first African American hired at Highland Hospital in Oakland, California, where he was assistant pharmacist. In 1942, |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 28, "sc": 365, "ep": 28, "ec": 1026} | 161,585 | Q7566320 | 28 | 365 | 28 | 1,026 | South Berkeley, Berkeley, California | Notable residents | while still working for the state, he purchased a pharmacy in Berkeley which he named Rumford's Pharmacy. In 1942, Berkeley Mayor Laurance L. Cross appointed Rumford to the Emergency Housing Committee, which sought to find housing for wartime laborers. In his capacity as committee member, he was able to push for more integrated housing. In his first year in the state assembly, Rumford succeeded in passing legislation barring discrimination in the state National Guard. One of Rumford's most important achievements was the passage of the 1959 Fair Employment Practices Act, which outlawed employment discrimination. In 1963, Rumford introduced assembly bill |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 28, "sc": 1026, "ep": 28, "ec": 1610} | 161,585 | Q7566320 | 28 | 1,026 | 28 | 1,610 | South Berkeley, Berkeley, California | Notable residents | 1240, the Fair Housing Bill. It became known as the Rumford Fair Housing Bill, and its purpose was to outlaw discrimination in housing. The bill was at the top of Governor Brown's legislative agenda, and it had been endorsed by the NAACP and the California Democratic Party. Nonetheless, it faced strong opposition and was amended several times before being passed by a vote of 47 to 24. When it reached the state senate, members of the Congress of Racial Equality occupied the rotunda of the California State Capitol. Rumford asked them to leave, but they refused. The bill was held |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 28, "sc": 1610, "ep": 28, "ec": 2237} | 161,585 | Q7566320 | 28 | 1,610 | 28 | 2,237 | South Berkeley, Berkeley, California | Notable residents | up for three months, and the committee didn't hold a hearing on it until the last day of the session. Despite the opposition of the California Real Estate Association, the Apartment House Owners Association, and the Chamber of Commerce, the bill passed the senate and was signed into law by Governor Brown.Rumford was honored at the 1972 World Symposium on Air Pollution Control, which recognized his contributions to the fight against air pollution.
In 1980, a segment of the Grove-Shafter Freeway was renamed the William Byron Rumford freeway in his honor. The postal station at the Oakland federal building is named |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 28, "sc": 2237, "ep": 28, "ec": 2863} | 161,585 | Q7566320 | 28 | 2,237 | 28 | 2,863 | South Berkeley, Berkeley, California | Notable residents | for him, as is a senior housing community in Berkeley. His archives are housed at the African American Museum and Library, a research center operated by the Oakland Public Library.
Mable Howard - (February 3, 1905 - March 29, 1994) - Community activist -
In the 1960s, the city of Berkeley planned to build new BART tracks that would run above ground through South Berkeley, where much of the city’s Black population is based. Howard, a local activist who moved from Galveston, Texas during World War II to work in the shipyards, feared BART would disrupt the neighborhood and tank property values.
So |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 28, "sc": 2863, "ep": 28, "ec": 3521} | 161,585 | Q7566320 | 28 | 2,863 | 28 | 3,521 | South Berkeley, Berkeley, California | Notable residents | Howard spearheaded litigation to halt the construction of the tracks for nine months, until BART agreed to build the line underground. A subsidized housing complex for seniors was later named after Howard for her contributions to South Berkeley. Howard is also the mother of artist Mildred Howard.
Mildred Howard - (1945) - visual artist - Mildred is an African-American artist known primarily for her sculptural installation and mixed-media assemblages. Her work has been shown at galleries in Boston, Los Angeles and New York, internationally at venues in Berlin, Cairo, London, Paris, and Venice, and at institutions including the Oakland Museum |
{"datasets_id": 161585, "wiki_id": "Q7566320", "sp": 28, "sc": 3521, "ep": 28, "ec": 3956} | 161,585 | Q7566320 | 28 | 3,521 | 28 | 3,956 | South Berkeley, Berkeley, California | Notable residents | of California, the de Young Museum, SFMOMA, the San Jose Museum of Art and the Museum of the African Diaspora. Howard was born in 1945 to Rolly and Mable Howard in San Francisco, California, and raised in South Berkeley, California. Howard has created numerous public installation works in the Bay Area and beyond. In 2017, a rent increase forced her to move out of the Berkeley, CA studio where she had lived and worked for 18 years. |
{"datasets_id": 161586, "wiki_id": "Q19879228", "sp": 2, "sc": 0, "ep": 10, "ec": 72} | 161,586 | Q19879228 | 2 | 0 | 10 | 72 | South East Football Netball League | History & Former Teams | South East Football Netball League History The competition had its origins in the South West Gippsland FL from 1954 to 1994. In 1995 the league was rolled in the MPNFL and the administrative duties were taken by the MPNFL management. While under the MPNFL control there were three minor re-distributions of clubs and that created different divisions, most of the clubs were in the MPNFL Northern Division 1995–98; MPNFL Peninsula Division 1999–2004;& MPNFL Casey-Cardinia League 2005–2014 Former Teams Hampton Park transferred to the Southern Football Netball League in 2018 |
{"datasets_id": 161587, "wiki_id": "Q4049874", "sp": 2, "sc": 0, "ep": 10, "ec": 93} | 161,587 | Q4049874 | 2 | 0 | 10 | 93 | Spec Ops II: Green Berets | Reception & Expansion | Spec Ops II: Green Berets Reception Spec Ops II: Green Berets received mixed to negative reviews. Aggregating review websites GameRankings and Metacritic gave the Microsoft Windows version 52.75% based on 20 reviews and the Dreamcast version 48.67% based on 3 reviews and 50/100 based on 8 reviews. Expansion Spec Ops II: Operation Bravo is an expansion pack and was released online the following year. |
{"datasets_id": 161588, "wiki_id": "Q357977", "sp": 2, "sc": 0, "ep": 10, "ec": 460} | 161,588 | Q357977 | 2 | 0 | 10 | 460 | Stephen Darby | Early life & Liverpool | Stephen Darby Early life Darby was brought up in Maghull, where he attended St. John Bosco's primary school and Maricourt Roman Catholic High School. Liverpool On 18 July 2006 Liverpool's official site announced that five academy players have been promoted to Melwood, Darby was one of them. Darby played in Liverpool's FA Youth Cup winning teams in 2006 and 2007 that defeated Manchester City and Manchester United's academies in the finals respectively, captaining the team in the 2006 final. He was first selected for the Liverpool squad which played Turkish side Galatasaray in the group stages of the UEFA Champions |
{"datasets_id": 161588, "wiki_id": "Q357977", "sp": 10, "sc": 460, "ep": 10, "ec": 1091} | 161,588 | Q357977 | 10 | 460 | 10 | 1,091 | Stephen Darby | Liverpool | League in December 2006, but he was an unused substitute in that match. Despite being on the first team bench he didn't forget his roots and after a four-hour flight from Istanbul to Liverpool he went straight to The Hawthorns and played a full game including extra-time to help the youth team through to the fourth round.
During 2007–08 season Darby was the captain of Liverpool's reserves team that ended the campaign as northern and national champions. His performances earned him praises from manager Gary Ablett calling him 'Mr. Consistency'.
His first appearance in a competitive match was as a substitute in |
{"datasets_id": 161588, "wiki_id": "Q357977", "sp": 10, "sc": 1091, "ep": 10, "ec": 1738} | 161,588 | Q357977 | 10 | 1,091 | 10 | 1,738 | Stephen Darby | Liverpool | Liverpool's League Cup fourth round defeat against Tottenham Hotspur in November 2008.
He made his Champions League debut against PSV Eindhoven on 9 December 2008, coming on as a substitute alongside fellow homegrown players Jay Spearing and Martin Kelly. On 5 July 2009, Darby secured a three-year extension to his contract along with fellow Melwood graduate Jay Spearing.
He made his first competitive start for Liverpool against ACF Fiorentina in the UEFA Champions League on 9 December 2009 in a 2–1 defeat. His second competitive start came against Reading in the FA Cup. The game finished 1–1. Darby was praised by Liverpool |
{"datasets_id": 161588, "wiki_id": "Q357977", "sp": 10, "sc": 1738, "ep": 14, "ec": 215} | 161,588 | Q357977 | 10 | 1,738 | 14 | 215 | Stephen Darby | Liverpool & Swindon | centre-back Jamie Carragher for his performance in the game.
Darby made his Premier League debut against Tottenham Hotspur on 20 January 2010, coming on as a substitute in the 90th minute for Philipp Degen.
He was included in Liverpool's 21 man squad for the 2010–11 Premier League season.
He was released by the club at the end of the 2011–12 season. Swindon On 11 March 2010, Darby joined Swindon Town on loan for the remainder of the 2009–10 season. He started his first game on 13 March 2010 against Brighton & Hove Albion and played the full 90 minutes at right back. He |
{"datasets_id": 161588, "wiki_id": "Q357977", "sp": 14, "sc": 215, "ep": 18, "ec": 421} | 161,588 | Q357977 | 14 | 215 | 18 | 421 | Stephen Darby | Swindon & Notts County | then took part in the play-off semi final against Charlton Athletic, where he scored the decisive penalty in the shoot-out. Notts County On 1 November 2010, Notts County manager and ex-Liverpool player Paul Ince snapped up both Darby and Thomas Ince on a loan deal from Liverpool. He made his County debut on 6 November in an FA Cup 1st round match against Gateshead On 13 November, he started in County's 3–1 league defeat to Exeter City. A week later, he started again in the 1–0 home defeat to Tranmere Rovers. On 23 November, he started and was named Man |
{"datasets_id": 161588, "wiki_id": "Q357977", "sp": 18, "sc": 421, "ep": 22, "ec": 182} | 161,588 | Q357977 | 18 | 421 | 22 | 182 | Stephen Darby | Notts County & Rochdale | of the Match in the 1–0 win against one of his former clubs, Swindon. Again he started on 11 December, this time in a win as County defeated the Milton Keynes Dons at home, 2–0. After his loan spell completed on 3 January 2011 he returned to Liverpool. He rejoined County later in January for the remainder of the season, before returning to Liverpool in early May. Rochdale On 7 July 2011, Rochdale confirmed the signing of Darby on loan for the 2011–12 campaign. He made his debut on 6 August 2011, against Sheffield Wednesday, playing the full 90 minutes |
{"datasets_id": 161588, "wiki_id": "Q357977", "sp": 22, "sc": 182, "ep": 26, "ec": 326} | 161,588 | Q357977 | 22 | 182 | 26 | 326 | Stephen Darby | Rochdale & Bradford City | in a 2–0 defeat. He was the fans' man of the match in the 0–0 home draw against Carlisle United on 16 August 2011. He made 40 appearances for the club during his season-long loan. Bradford City On 4 July 2012, Darby signed for Bradford City on a two-year contract. He made his debut on 11 August 2012 in a 1–0 League Cup win against Notts County. He made his league debut a week later against Gillingham. He made his home debut on 21 August, coming on as a substitute for Nahki Wells in a 1–0 win against Fleetwood Town. |
{"datasets_id": 161588, "wiki_id": "Q357977", "sp": 26, "sc": 326, "ep": 26, "ec": 916} | 161,588 | Q357977 | 26 | 326 | 26 | 916 | Stephen Darby | Bradford City | On 25 September, he scored the only goal of his career in the 115th minute of a 3–2 win against Burton Albion in the League Cup third round. Bradford went on to reach the final, beating Premier League sides Wigan Athletic, Arsenal and Aston Villa in subsequent rounds.
Darby won 7 awards at the Bradford City end-of-season dinner in May 2014. Darby was made club captain after Gary Jones joined Notts County, and signed a new three-year contract in June 2014. For the 2016–17 season, Darby was replaced by new signing Romain Vincelot as captain, under new manager Stuart McCall. In |
{"datasets_id": 161588, "wiki_id": "Q357977", "sp": 26, "sc": 916, "ep": 38, "ec": 210} | 161,588 | Q357977 | 26 | 916 | 38 | 210 | Stephen Darby | Bradford City & Bolton Wanderers & Retirement & Personal life | May 2017 it was announced that he would be released at the end of the season, when his contract expired. Bolton Wanderers On 7 July 2017, Darby signed a two-year contract at Bolton Wanderers, reuniting with former manager Phil Parkinson. Retirement On 18 September 2018, Darby announced his retirement from professional football at the age of 29 after being diagnosed with motor neuron disease. Personal life Darby is married to Steph Houghton, captain of Manchester City Women and the England women's national team. They married on 21 June 2018.
Darby set up the Darby Rimmer MND Foundation to fund support for |
{"datasets_id": 161588, "wiki_id": "Q357977", "sp": 38, "sc": 210, "ep": 42, "ec": 197} | 161,588 | Q357977 | 38 | 210 | 42 | 197 | Stephen Darby | Personal life & International career | families affected by the disease, and research a cure. In July 2019 two of his former clubs (Liverpool and Bradford City) held a fundraising match. International career Darby took part in England's squad for UEFA European U19 Championship qualifications in May 2007 and made two appearances alongside Liverpool teammates Jack Hobbs, Craig Lindfield and Adam Hammill. |
{"datasets_id": 161589, "wiki_id": "Q92774", "sp": 2, "sc": 0, "ep": 6, "ec": 621} | 161,589 | Q92774 | 2 | 0 | 6 | 621 | Stephen R. Bourne | Biography | Stephen R. Bourne Biography Bourne has a bachelor's degree in mathematics from King's College London, England. He has a Diploma in Computer Science and a Ph.D. in mathematics from Trinity College, Cambridge. Subsequently he worked on an ALGOL 68 compiler at the University of Cambridge Computer Laboratory (see ALGOL 68C).
After Cambridge, Bourne spent nine years at Bell Labs with the Seventh Edition Unix team. As well as the Bourne shell, he wrote the adb debugger and The UNIX System, the second book on the UNIX system, intended for a general readership.
After Bell Labs, Bourne worked in senior engineering management positions |
{"datasets_id": 161589, "wiki_id": "Q92774", "sp": 6, "sc": 621, "ep": 6, "ec": 1248} | 161,589 | Q92774 | 6 | 621 | 6 | 1,248 | Stephen R. Bourne | Biography | at Silicon Graphics, Digital Equipment Corporation, Sun Microsystems and Cisco Systems.
From 2000 to 2002 he was president of the Association for Computing Machinery. For his work on computing, Bourne was awarded the ACM's Presidential Award in 2008 and was made a fellow of the organization in 2005. He is also a Fellow of the Royal Astronomical Society.
Bourne was chief technology officer at Icon Venture Partners, a venture capital firm based in Menlo Park, California through 2014. He is also the chair of the editorial advisory board for ACM Queue, a magazine he helped found when he was president of the |
{"datasets_id": 161589, "wiki_id": "Q92774", "sp": 6, "sc": 1248, "ep": 6, "ec": 1253} | 161,589 | Q92774 | 6 | 1,248 | 6 | 1,253 | Stephen R. Bourne | Biography | ACM. |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 2, "sc": 0, "ep": 4, "ec": 629} | 161,590 | Q176737 | 2 | 0 | 4 | 629 | Stochastic process | Stochastic process In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to |
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{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 4, "sc": 629, "ep": 4, "ec": 1353} | 161,590 | Q176737 | 4 | 629 | 4 | 1,353 | Stochastic process | vary in a random manner. They have applications in many disciplines including sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.
Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the |
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{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 4, "sc": 1353, "ep": 4, "ec": 1993} | 161,590 | Q176737 | 4 | 1,353 | 4 | 1,993 | Stochastic process | Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.
The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific |
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{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 4, "sc": 1993, "ep": 4, "ec": 2658} | 161,590 | Q176737 | 4 | 1,993 | 4 | 2,658 | Stochastic process | mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.
Based on their mathematical properties, stochastic processes can be divided into various categories, which include random walks, martingales, Markov processes, Lévy processes, |
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{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 4, "sc": 2658, "ep": 8, "ec": 136} | 161,590 | Q176737 | 4 | 2,658 | 8 | 136 | Stochastic process | Introduction | Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications. Introduction A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 8, "sc": 136, "ep": 8, "ec": 731} | 161,590 | Q176737 | 8 | 136 | 8 | 731 | Stochastic process | Introduction | each random variable of the stochastic process is uniquely associated with an element in the set. The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. Each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real line or -dimensional Euclidean space. An increment is the amount that a stochastic process changes between two index |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 8, "sc": 731, "ep": 12, "ec": 370} | 161,590 | Q176737 | 8 | 731 | 12 | 370 | Stochastic process | Introduction & Classifications | values, often interpreted as two points in time. A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization. Classifications A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space.
When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 12, "sc": 370, "ep": 12, "ec": 1009} | 161,590 | Q176737 | 12 | 370 | 12 | 1,009 | Stochastic process | Classifications | as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time. If the index set is some interval of the real line, then time is said to be continuous. The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable. If the index set is the integers, or some subset of them, then the stochastic |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 12, "sc": 1009, "ep": 16, "ec": 142} | 161,590 | Q176737 | 12 | 1,009 | 16 | 142 | Stochastic process | Classifications & Etymology | process can also be called a random sequence.
If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is -dimensional Euclidean space, then the stochastic process is called a -dimensional vector process or -vector process. Etymology The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 16, "sc": 142, "ep": 16, "ec": 753} | 161,590 | Q176737 | 16 | 142 | 16 | 753 | Stochastic process | Etymology | word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence. In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz who in 1917 wrote in German the word stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. For the term and a specific |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 16, "sc": 753, "ep": 16, "ec": 1390} | 161,590 | Q176737 | 16 | 753 | 16 | 1,390 | Stochastic process | Etymology | mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin, though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.
Early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 16, "sc": 1390, "ep": 20, "ec": 384} | 161,590 | Q176737 | 16 | 1,390 | 20 | 384 | Stochastic process | Etymology & Terminology | French verb meaning "to run" or "to gallop". The first written appearance of the term random process pre-dates stochastic process, which the Oxford English Dictionary also gives as a synonym, and was used in an article by Francis Edgeworth published in 1888. Terminology The definition of a stochastic process varies, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. The terms random process and stochastic process are considered synonyms and are used interchangeably, without the index set being precisely specified. Both "collection", or "family" are used while instead of "index set", sometimes |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 20, "sc": 384, "ep": 24, "ec": 42} | 161,590 | Q176737 | 20 | 384 | 24 | 42 | Stochastic process | Terminology & Notation | the terms "parameter set" or "parameter space" are used.
The term random function is also used to refer to a stochastic or random process, though sometimes it is only used when the stochastic process takes real values. This term is also used when the index sets are mathematical spaces other than the real line, while the terms stochastic process and random process are usually used when the index set interpreted as time, and other terms are used such as random field when the index set is -dimensional Euclidean space or a manifold. Notation A stochastic process can be denoted, among |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 24, "sc": 42, "ep": 28, "ec": 224} | 161,590 | Q176737 | 24 | 42 | 28 | 224 | Stochastic process | Notation & Bernoulli process | other ways, by , , or simply as or , although is regarded as an abuse of notation. For example, or are used to refer to the random variable with the index , and not the entire stochastic process. If the index set is , then one can write, for example, to denote the stochastic process. Bernoulli process One of the simplest stochastic processes is the Bernoulli process, which is a sequence of independent and identically distributed (iid) random variables, where each random variable takes either the value one or zero, say one |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 28, "sc": 224, "ep": 32, "ec": 198} | 161,590 | Q176737 | 28 | 224 | 32 | 198 | Stochastic process | Bernoulli process & Random walk | with probability and zero with probability . This process can be linked to repeatedly flipping a coin, where the probability of obtaining a head is and its value is one, while the value of a tail is zero. In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, where each coin flip is an example of a Bernoulli trial. Random walk Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. But some also |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 32, "sc": 198, "ep": 32, "ec": 800} | 161,590 | Q176737 | 32 | 198 | 32 | 800 | Stochastic process | Random walk | use the term to refer to processes that change in continuous time, particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism. There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.
A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 32, "sc": 800, "ep": 36, "ec": 163} | 161,590 | Q176737 | 32 | 800 | 36 | 163 | Stochastic process | Random walk & Wiener process | process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, , or decreases by one with probability , so index set of this random walk is the natural numbers, while its state space is the integers. If the , this random walk is called a symmetric random walk. Wiener process The Wiener process is a stochastic process with stationary and independent increments that are normally distributed based on the size of the increments. The Wiener |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 36, "sc": 163, "ep": 36, "ec": 805} | 161,590 | Q176737 | 36 | 163 | 36 | 805 | Stochastic process | Wiener process | process is named after Norbert Wiener, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for Brownian movement in liquids.
Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. But the process can be defined more generally so its state |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 36, "sc": 805, "ep": 36, "ec": 1392} | 161,590 | Q176737 | 36 | 805 | 36 | 1,392 | Stochastic process | Wiener process | space can be -dimensional Euclidean space. If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant , which is a real number, then the resulting stochastic process is said to have drift .
Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk. The process arises as the mathematical |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 36, "sc": 1392, "ep": 36, "ec": 2062} | 161,590 | Q176737 | 36 | 1,392 | 36 | 2,062 | Stochastic process | Wiener process | limit of other stochastic processes such as certain random walks rescaled, which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.
The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes. The process also has many applications and is the main stochastic process used in stochastic calculus. It plays a central role in quantitative finance, where it is used, for example, in the Black–Scholes–Merton model. The process is also used in different fields, including the majority of natural sciences as well as |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 36, "sc": 2062, "ep": 40, "ec": 485} | 161,590 | Q176737 | 36 | 2,062 | 40 | 485 | Stochastic process | Wiener process & Poisson process | some branches of social sciences, as a mathematical model for various random phenomena. Poisson process The Poisson process is a stochastic process that has different forms and definitions. It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 40, "sc": 485, "ep": 40, "ec": 1119} | 161,590 | Q176737 | 40 | 485 | 40 | 1,119 | Stochastic process | Poisson process | as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.
If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes.
The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. If |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 40, "sc": 1119, "ep": 40, "ec": 1781} | 161,590 | Q176737 | 40 | 1,119 | 40 | 1,781 | Stochastic process | Poisson process | the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.
Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects. But then it can be defined on the -dimensional Euclidean |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 40, "sc": 1781, "ep": 44, "ec": 57} | 161,590 | Q176737 | 40 | 1,781 | 44 | 57 | Stochastic process | Poisson process & Stochastic process | space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. In this setting, the Poisson process, also called the Poisson point process, is one of the most important objects in probability theory, both for applications and theoretical reasons. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces. Stochastic process A stochastic process is defined as a collection of random |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 44, "sc": 57, "ep": 44, "ec": 620} | 161,590 | Q176737 | 44 | 57 | 44 | 620 | Stochastic process | Stochastic process | variables defined on a common probability space , where is a sample space, is a -algebra, and is a probability measure; and the random variables, indexed by some set , all take values in the same mathematical space , which must be measurable with respect to some -algebra .
In other words, for a given probability space and a measurable space , a stochastic process is a collection of -valued random variables, which can be written as:
Historically, in many problems from the natural sciences a point had the meaning of time, so is a random |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 44, "sc": 620, "ep": 48, "ec": 52} | 161,590 | Q176737 | 44 | 620 | 48 | 52 | Stochastic process | Stochastic process & Index set | variable representing a value observed at time . A stochastic process can also be written as to reflect that it is actually a function of two variables, and .
There are others ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a -valued random variable, where is the space of all the possible -valued functions of that map from the set into the space . Index set The set is called the index set or parameter set of |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 48, "sc": 52, "ep": 52, "ec": 61} | 161,590 | Q176737 | 48 | 52 | 52 | 61 | Stochastic process | Index set & State space | the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set the interpretation of time. In addition to these sets, the index set can be other linearly ordered sets or more general mathematical sets, such as the Cartesian plane or -dimensional Euclidean space, where an element can represent a point in space. But in general more results and theorems are possible for stochastic processes when the index set is ordered. State space The mathematical space of a stochastic process is called its |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 52, "sc": 61, "ep": 56, "ec": 340} | 161,590 | Q176737 | 52 | 61 | 56 | 340 | Stochastic process | State space & Sample function | state space. This mathematical space can be defined using integers, real lines, -dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take. Sample function A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if is a stochastic process, then for any point , the mapping
is called a sample function, a realization, or, particularly when is interpreted as time, |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 56, "sc": 340, "ep": 60, "ec": 297} | 161,590 | Q176737 | 56 | 340 | 60 | 297 | Stochastic process | Sample function & Increment | a sample path of the stochastic process . This means that for a fixed , there exists a sample function that maps the index set to the state space . Other names for a sample function of a stochastic process include trajectory, path function or path. Increment An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if is a |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 60, "sc": 297, "ep": 64, "ec": 185} | 161,590 | Q176737 | 60 | 297 | 64 | 185 | Stochastic process | Increment & Law | stochastic process with state space and index set , then for any two non-negative numbers and such that , the difference is a -valued random variable known as an increment. When interested in the increments, often the state space is the real line or the natural numbers, but it can be -dimensional Euclidean space or more abstract spaces such as Banach spaces. Law For a stochastic process defined on the probability space , the law of stochastic process is defined as the image measure:
where is a probability measure, the symbol denotes |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 64, "sc": 185, "ep": 68, "ec": 104} | 161,590 | Q176737 | 64 | 185 | 68 | 104 | Stochastic process | Law & Finite-dimensional probability distributions | function composition and is the pre-image of the measurable function or, equivalently, the -valued random variable , where is the space of all the possible -valued functions of , so the law of a stochastic process is a probability measure.
For a measurable subset of , the pre-image of gives
so the law of a can be written as:
The law of a stochastic process or a random variable is also called the probability law, probability distribution, or the distribution. Finite-dimensional probability distributions For a stochastic process with law , its finite-dimensional distributions are defined as:
where |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 68, "sc": 104, "ep": 72, "ec": 175} | 161,590 | Q176737 | 68 | 104 | 72 | 175 | Stochastic process | Finite-dimensional probability distributions & Stationarity | is a counting number and each set is a non-empty finite subset of the index set , so each , which means that is any finite collection of subsets of the index set .
For any measurable subset of the -fold Cartesian power , the finite-dimensional distributions of a stochastic process can be written as:
The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions. Stationarity Stationarity is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 72, "sc": 175, "ep": 72, "ec": 760} | 161,590 | Q176737 | 72 | 175 | 72 | 760 | Stochastic process | Stationarity | is a stationary stochastic process, then for any the random variable has the same distribution, which means that for any set of index set values , the corresponding random variables
all have the same probability distribution. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line. But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.
When the index set can be interpreted as time, a stochastic process is said to be |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 72, "sc": 760, "ep": 72, "ec": 1449} | 161,590 | Q176737 | 72 | 760 | 72 | 1,449 | Stochastic process | Stationarity | stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations. The intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed.
A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. One example is when a discrete-time or |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 72, "sc": 1449, "ep": 76, "ec": 175} | 161,590 | Q176737 | 72 | 1,449 | 76 | 175 | Stochastic process | Stationarity & Filtration | continuous-time stochastic process is said to be stationary in the wide sense, then the process has a finite second moment for all and the covariance of the two random variables and depends only on the number for all . Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense. Filtration A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such in the case of |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 76, "sc": 175, "ep": 76, "ec": 678} | 161,590 | Q176737 | 76 | 175 | 76 | 678 | Stochastic process | Filtration | the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration , on a probability space is a family of sigma-algebras such that for all , where and denotes the total order of the index set . With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process at , which can be interpreted as time . The intuition behind a filtration is that as time passes, more |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 76, "sc": 678, "ep": 80, "ec": 442} | 161,590 | Q176737 | 76 | 678 | 80 | 442 | Stochastic process | Filtration & Modification | and more information on is known or available, which is captured in , resulting in finer and finer partitions of . Modification A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process that has the same index set , set space , and probability space as another stochastic process is said to be a modification of if for all the following
holds. Two stochastic processes that are modifications of each other have the same law and they are said to be |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 80, "sc": 442, "ep": 80, "ec": 1146} | 161,590 | Q176737 | 80 | 442 | 80 | 1,146 | Stochastic process | Modification | stochastically equivalent or equivalent.
Instead of modification, the term version is also used, however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.
If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 80, "sc": 1146, "ep": 88, "ec": 73} | 161,590 | Q176737 | 80 | 1,146 | 88 | 73 | Stochastic process | Modification & Indistinguishable & Separability | continuous modification or version. The theorem can also be generalized to random fields so the index set is -dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces. Indistinguishable Two stochastic processes and defined on the same probability space with the same index set and set space are said be indistinguishable if the following
holds. If two and are modifications of each other and are almost surely continuous, then and are indistinguishable. Separability Separability is a property of a stochastic process based on its index set |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 88, "sc": 73, "ep": 88, "ec": 666} | 161,590 | Q176737 | 88 | 73 | 88 | 666 | Stochastic process | Separability | 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, which means that the index set has a dense countable subset.
More precisely, a real-valued continuous-time stochastic process with a probability space is separable if its index set has a dense countable subset and there is a set of probability zero, so , such that for every open set |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 88, "sc": 666, "ep": 88, "ec": 1231} | 161,590 | Q176737 | 88 | 666 | 88 | 1,231 | Stochastic process | Separability | and every closed set , the two events and differ from each other at most on a subset of .
The definition of separability can also be stated for other index sets and state spaces, such as in the case of random fields, where the index set as well as the state space can be -dimensional Euclidean space.
The concept of separability of a stochastic process was introduced by Joseph Doob, where the underlying idea is to make a countable set of points of the index set determine the properties of the stochastic process. Any stochastic process with a |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 88, "sc": 1231, "ep": 92, "ec": 192} | 161,590 | Q176737 | 88 | 1,231 | 92 | 192 | Stochastic process | Separability & Independence | countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line. Independence Two stochastic processes and defined on the same probability space with the same index set are said be independent if for all and for every choice of epochs , the random vectors and are |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 92, "sc": 192, "ep": 100, "ec": 455} | 161,590 | Q176737 | 92 | 192 | 100 | 455 | Stochastic process | Independence & Independence implies uncorrelatedness & Skorokhod space | independent. Independence implies uncorrelatedness If two stochastic processes and are independent, then they are also uncorrelated. Skorokhod space A Skorokhod space, also written as Skorohod space, is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as or , and take values on the real line or on some metric space. Such functions are known as càdlàg or cadlag functions, based on the acronym of the French expression continue à droite, limite à gauche, due to the functions being right-continuous with left limits. A |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 100, "sc": 455, "ep": 100, "ec": 1101} | 161,590 | Q176737 | 100 | 455 | 100 | 1,101 | Stochastic process | Skorokhod space | Skorokhod function space, introduced by Anatoliy Skorokhod, is often denoted with the letter , so the function space is also referred to as space . The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, denotes the space of càdlàg functions defined on the unit interval .
Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space. Such spaces contain continuous functions, which correspond to sample functions of the |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 100, "sc": 1101, "ep": 106, "ec": 20} | 161,590 | Q176737 | 100 | 1,101 | 106 | 20 | Stochastic process | Skorokhod space & Regularity & Markov processes and chains | Wiener process. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process (on the real line), are also members of this space. Regularity In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues. For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous. Markov processes and |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 106, "sc": 20, "ep": 108, "ec": 636} | 161,590 | Q176737 | 106 | 20 | 108 | 636 | Stochastic process | Markov processes and chains | chains Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.
The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes in continuous time, while random walks on the integers and the gambler's ruin |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 108, "sc": 636, "ep": 108, "ec": 1228} | 161,590 | Q176737 | 108 | 636 | 108 | 1,228 | Stochastic process | Markov processes and chains | problem are examples of Markov processes in discrete time.
A Markov chain is a type of Markov process that has either discrete state space or discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space). It |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 108, "sc": 1228, "ep": 108, "ec": 1893} | 161,590 | Q176737 | 108 | 1,228 | 108 | 1,893 | Stochastic process | Markov processes and chains | has been argued that the first definition of a Markov chain, where it has discrete time, now tends to be used, despite the second definition having been used by researchers like Joseph Doob and Kai Lai Chung.
Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.
The concept of the Markov property was originally for stochastic processes in continuous and |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 108, "sc": 1893, "ep": 112, "ec": 434} | 161,590 | Q176737 | 108 | 1,893 | 112 | 434 | Stochastic process | Markov processes and chains & Martingale | discrete time, but the property has been adapted for other index sets such as -dimensional Euclidean space, which results in collections of random variables known as Markov random fields. Martingale A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 112, "sc": 434, "ep": 112, "ec": 1116} | 161,590 | Q176737 | 112 | 434 | 112 | 1,116 | Stochastic process | Martingale | coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, but they can also be complex-valued or even more general.
A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. For a sequence of independent and identically distributed random variables with zero mean, the stochastic process formed from the successive partial sums is a discrete-time martingale. In this aspect, discrete-time martingales generalize the idea of partial sums of independent |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 112, "sc": 1116, "ep": 112, "ec": 1786} | 161,590 | Q176737 | 112 | 1,116 | 112 | 1,786 | Stochastic process | Martingale | random variables.
Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the compensated Poisson process. Martingales can also be built from other martingales. For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.
Martingales mathematically formalize the idea of a fair game, and they were originally developed to show that it is not possible to win a fair game. But now they are used in many areas of probability, which is one of the main |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 112, "sc": 1786, "ep": 112, "ec": 2447} | 161,590 | Q176737 | 112 | 1,786 | 112 | 2,447 | Stochastic process | Martingale | reasons for studying them. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.
Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance. |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 114, "sc": 0, "ep": 116, "ec": 667} | 161,590 | Q176737 | 114 | 0 | 116 | 667 | Stochastic process | Lévy process | Lévy process Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process is a Lévy process if for non-negatives numbers, , the corresponding increments
are all independent of each other, and the distribution of each increment only depends on the difference in time.
A Lévy |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 116, "sc": 667, "ep": 120, "ec": 169} | 161,590 | Q176737 | 116 | 667 | 120 | 169 | Stochastic process | Lévy process & Random field | process can be defined such that its state space is some abstract mathematical space, such as a Banach space, but the processes are often defined so that they take values in Euclidean space. The index set is the non-negative numbers, so , which gives the interpretation of time. Important stochastic processes such as the Wiener process, the homogeneous Poisson process (in one dimension), and subordinators are all Lévy processes. Random field A random field is a collection of random variables indexed by a -dimensional Euclidean space or some manifold. In general, a random field can be considered an example of |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 120, "sc": 169, "ep": 124, "ec": 138} | 161,590 | Q176737 | 120 | 169 | 124 | 138 | Stochastic process | Random field & Point process | a stochastic or random process, where the index set is not necessarily a subset of the real line. But there is a convention that an indexed collection of random variables is called a random field when the index has two or more dimensions. If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process. Point process A point process is a collection of points randomly located on some mathematical space such as the real line, -dimensional Euclidean space, |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 124, "sc": 138, "ep": 124, "ec": 742} | 161,590 | Q176737 | 124 | 138 | 124 | 742 | Stochastic process | Point process | or more abstract spaces. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 124, "sc": 742, "ep": 128, "ec": 248} | 161,590 | Q176737 | 124 | 742 | 128 | 248 | Stochastic process | Point process & Early probability theory | not clear.
Other authors consider a point process as a stochastic process, where the process is indexed by sets of the underlying space on which it is defined, such as the real line or -dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes. Early probability theory Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago, but very little analysis on them was done in terms of probability. The year 1654 is often considered the birth of |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 128, "sc": 248, "ep": 128, "ec": 964} | 161,590 | Q176737 | 128 | 248 | 128 | 964 | Stochastic process | Early probability theory | probability theory when French mathematicians Pierre Fermat and Blaise Pascal had a written correspondence on probability, motivated by a gambling problem. But there was earlier mathematical work done on the probability of gambling games such as Liber de Ludo Aleae by Gerolamo Cardano, written in the 16th century but posthumously published later in 1663.
After Cardano, Jakob Bernoulli wrote Ars Conjectandi, which is considered a significant event in the history of probability theory. Bernoulli's book was published, also posthumously, in 1713 and inspired many mathematicians to study probability. But despite some renown mathematicians contributing to probability theory, such as Pierre-Simon Laplace, |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 128, "sc": 964, "ep": 132, "ec": 477} | 161,590 | Q176737 | 128 | 964 | 132 | 477 | Stochastic process | Early probability theory & Statistical mechanics | Abraham de Moivre, Carl Gauss, Siméon Poisson and Pafnuty Chebyshev, most of the mathematical community did not consider probability theory to be part of mathematics until the 20th century. Statistical mechanics In the physical sciences, scientists developed in the 19th century the discipline of statistical mechanics, where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as Rudolf Clausius, most of the work had little or no randomness.
This changed in 1859 when James Clerk Maxwell |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 132, "sc": 477, "ep": 136, "ec": 138} | 161,590 | Q176737 | 132 | 477 | 136 | 138 | Stochastic process | Statistical mechanics & Measure theory and probability theory | contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities. The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, Ludwig Boltzmann and Josiah Gibbs, which would later have an influence on Albert Einstein's mathematical model for Brownian movement. Measure theory and probability theory At the International Congress of Mathematicians in Paris in 1900, David Hilbert presented a list of mathematical problems, where his sixth |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 136, "sc": 138, "ep": 136, "ec": 862} | 161,590 | Q176737 | 136 | 138 | 136 | 862 | Stochastic process | Measure theory and probability theory | problem asked for a mathematical treatment of physics and probability involving axioms. 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 published the first probability book that used ideas from measure theory.
In 1920s fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as Sergei Bernstein, Aleksandr Khinchin, and Andrei Kolmogorov. Kolmogorov published in 1929 his first attempt at presenting a mathematical foundation, based |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 136, "sc": 862, "ep": 140, "ec": 308} | 161,590 | Q176737 | 136 | 862 | 140 | 308 | Stochastic process | Measure theory and probability theory & Birth of modern probability theory | on measure theory, for probability theory. In the early 1930s Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as Eugene Slutsky and Nikolai Smirnov, and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line. Birth of modern probability theory In 1933 Andrei Kolmogorov published in German his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung, where Kolmogorov used measure theory to develop an axiomatic framework for probability theory. The publication of this book is now widely considered to be the |
{"datasets_id": 161590, "wiki_id": "Q176737", "sp": 140, "sc": 308, "ep": 140, "ec": 1014} | 161,590 | Q176737 | 140 | 308 | 140 | 1,014 | Stochastic process | Birth of modern probability theory | birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics.
After the publication of Kolmogorov's book, further fundamental work on probability theory and stochastic processes was done by Khinchin and Kolmogorov as well as other mathematicians such as Joseph Doob, William Feller, Maurice Fréchet, Paul Lévy, Wolfgang Doeblin, and Harald Cramér.
Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory". World War II greatly interrupted the development of probability theory, causing, for example, the migration of Feller from Sweden to the United States of America and the death of |
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