|
Marcinkiewicz–Zygmund Inequality
In mathematics, the Marcinkiewicz–Zygmund inequality, named after Józef Marcinkiewicz and Antoni Zygmund, gives relations between moments of a collection of independent random variables. It is a generalization of the rule for the sum of variances of independent random variables to moments of arbitrary order. It is a special case of the Burkholder-Davis-Gundy inequality in the case of discrete-time martingales. Statement of the inequality Theorem J. Marcinkiewicz and A. Zygmund. Sur les fonctions indépendantes. ''Fund. Math.'', 28:60–90, 1937. Reprinted in Józef Marcinkiewicz, ''Collected papers'', edited by Antoni Zygmund, Panstwowe Wydawnictwo Naukowe, Warsaw, 1964, pp. 233–259. Yuan Shih Chow and Henry Teicher. ''Probability theory. Independence, interchangeability, martingales''. Springer-Verlag, New York, second edition, 1988. If \textstyle X_, \textstyle i=1,\ldots,n, are independent random variables such that \textstyle E\left( X_\right) =0 and \text ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Józef Marcinkiewicz
Józef Marcinkiewicz (; 30 March 1910 in Cimoszka, near Białystok, Poland – 1940 in Katyn, USSR) was a Polish mathematician. He was a student of Antoni Zygmund; and later worked with Juliusz Schauder, Stefan Kaczmarz and Raphaël Salem. He was a professor of the Stefan Batory University in Wilno. He enlisted in the Polish Army during the German invasion of Poland. In the aftermath of the simultaneous Soviet invasion of Poland, Marcinkiewicz was taken as a Polish POW to a Soviet camp in Starobielsk. The exact place and date of his death remain unknown, but it is believed that he died, when aged 30, in the Katyn massacre on the mass murder site near Smolensk. His parents, to whom he gave his manuscripts before the beginning of World War II, were transported to the Soviet Union in 1940 and later died of hunger in a camp. Their fate is described by ZygmundSee his commemoration in the volume of Marcinkiewicz's collected papers . described the last and his lost mathematical w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antoni Zygmund
Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. Zygmund was responsible for creating the Chicago school of mathematical analysis together with his doctoral student Alberto Calderón, for which he was awarded the National Medal of Science in 1986. Biography Born in Warsaw, Zygmund obtained his Ph.D. from the University of Warsaw (1923) and was a professor at Stefan Batory University at Wilno from 1930 to 1939, when World War II broke out and Poland was occupied. In 1940 he managed to emigrate to the United States, where he became a professor at Mount Holyoke College in South Hadley, Massachusetts. In 1945–1947 he was a professor at the University of Pennsylvania, and from 1947, until his retirement, at the University of Chicago. He was a member of several scientific societies. Fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Random Variables
Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independents (Oporto artist group), a Portuguese artist group historically linked to abstract art and to Fernando Lanhas, the central figure of Portuguese abstractionism Music Groups, labels, and genres * Independent music, a number of genres associated with independent labels * Independent record label, a record label not associated with a major label * Independent Albums, American albums chart Albums * ''Independent'' (Ai album), 2012 * ''Independent'' (Faze album), 2006 * ''Independent'' (Sacred Reich album), 1993 Songs * "Independent" (song), a 2007 song by Webbie * "Independent", a 2002 song by Ayumi Hamasaki from '' H'' News and media organizations * ''The Independent'', a British online newspaper. * ''The Malta Independent'', a Mal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burkholder-Davis-Gundy Inequality
In mathematics, quadratic variation is used in the analysis of 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. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...es such as Wiener process, Brownian motion and other Martingale (probability theory), martingales. Quadratic variation is just one kind of Total variation, variation of a process. Definition Suppose that X_t is a real-valued stochastic process defined on a probability space (\Omega,\mathcal,\mathbb) and with time index t ranging over the non-negative real numbers. Its quadratic variation is the process, written as [X]_t, defined as :[X]_t=\lim_\sum_^n(X_-X_)^2 where P ranges over partition of an interval, partitions of the interval [0,t] and the norm of the partition P is the mesh (mathematics), mesh. This limit, if it exists, is defined using Converge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Khintchine Inequality
In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick N complex numbers x_1,\dots,x_N \in\mathbb, and add them together each multiplied by a random sign \pm 1 , then the expected value of the sum's modulus, or the modulus it will be closest to on average, will be not too far off from \sqrt. Statement Let \_^N be i.i.d. random variables with P(\varepsilon_n=\pm1)=\frac12 for n=1,\ldots, N, i.e., a sequence with Rademacher distribution. Let 0 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rosenthal Inequalities
Rosenthal is a German and Jewish surname meaning "rose valley". Notable people with the name include: A * Abe M. Rosenthal (1922–2006), ''New York Times'' editor and columnist *Albert Rosenthal (1863–1939), American portrait artist *Albert J. Rosenthal (1919–2010), American legal scholar *Amy Krouse Rosenthal (1965-2017), American author *Arnold Jack Rosenthal (1923–2010), Louisiana politician; see Peter Beer *Arthur Rosenthal (1887–1959), German mathematician B *Barbara Rosenthal (born 1948), artist and writer *Ben Rosenthal (baseball) (born 1979), American baseball coach *Benjamin Stanley Rosenthal, United States Congressman (1962–1983) * Bernard "Tony" Rosenthal (1914–2009), American abstract sculptor * Bernard Rosenthal (scholar), American scholar of the Salem witchcraft trials C *Sir Charles Rosenthal (1875–1954), Australian general of World War I * Chuck Rosenthal (district attorney), Republican District Attorney of Harris County, Texas *Constantin Da ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypothesis. The average (or mean) of sample values is a statistic. The term statistic is used both for the function and for the value of the function on a given sample. When a statistic is being used for a specific purpose, it may be referred to by a name indicating its purpose. When a statistic is used for estimating a population parameter, the statistic is called an ''estimator''. A population parameter is any characteristic of a population under study, but when it is not feasible to directly measure the value of a population parameter, statistical methods are used to infer the likely value of the parameter on the basis of a statistic computed from a sample taken from the population. For example, the sample mean is an unbiased estimator of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Inequalities
Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An expe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Inequalities
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Probabilistic Inequalities")