Ibragimov–Iosifescu Conjecture For φ-mixing Sequences

	Ibragimov–Iosifescu Conjecture For φ-mixing Sequences Ibragimov–Iosifescu conjecture for φ-mixing sequences in probability theory is the collective name for 2 closely-related conjectures by Ildar Ibragimov and :ro:Marius Iosifescu. Conjecture Let (X_n,n\in\Bbb N) be a strictly stationary \phi-mixing sequence, for which \mathbb E(X_0^2)<\infty and $\operatorname(S_n)\to +\infty$ . Then $S_n:=\sum_^nX_j$ is asymptotically normally distributed. $\phi$ -mixing coefficients are defined as $\phi_X(n):=\sup(, \mu(B\mid A)-\mu(B), , A\in\mathcal F^m, B\in \mathcal F_,m\in\Bbb N )$ , where $\mathcal F^m$ and $\mathcal F_$ are the $\sigma$ -algebras generated by the $X_j, j\leqslant m$ (respectively $j\geqslant m+n$ ), and $\phi$ -mixing means that $\phi_X(n)\to 0$ . Reformulated: Suppose $X:=(X_k, k \in )$ is a strictly stationary sequence of random variables such that $EX_0 = 0, \ EX_0^2 < \infty$ [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Probability Theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Magda Peligrad Magda Peligrad is a Romanian mathematician and mathematical statistician known for her research in probability theory, and particularly on central limit theorems and stochastic processes. She works at the University of Cincinnati, where she is Distinguished Charles Phelps Taft Professor of Mathematical Sciences. Education and career Peligrad obtained her Ph.D. in 1980 from the Center of Statistics of the Romanian Academy. By 1983 she was working at the Sapienza University of Rome and by 1984 she had arrived at Cincinnati, where since 1988 she has supervised the dissertations of seven doctoral students. With Florence Merlevède and Sergey Utev, she is coauthor of the book ''Functional Gaussian Approximation for Dependent Structures'' (Oxford University Press, 2019). Recognition In 1995, Peligrad was elected as a Fellow of the Institute of Mathematical Statistics The Institute of Mathematical Statistics is an international professional and scholarly society devoted to the develo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Yuri Linnik Yuri Vladimirovich Linnik (russian: Ю́рий Влади́мирович Ли́нник; January 8, 1915 – June 30, 1972) was a Soviet mathematician active in number theory, probability theory and mathematical statistics. Linnik was born in Bila Tserkva, in present-day Ukraine. He went to St Petersburg University where his supervisor was Vladimir Tartakovski, and later worked at that university and the Steklov Institute. He was a member of the Russian Academy of Sciences, as was his father, Vladimir Pavlovich Linnik. He was awarded both State and Lenin Prizes. He died in Leningrad. Work in number theory * Linnik's theorem in analytic number theory * The dispersion method (which allowed him to solve the Titchmarsh problem). * The large sieve (which turned out to be extremely influential). * An elementary proof of the Hilbert-Waring theorem; see also Schnirelmann density. * The Linnik ergodic method, see , which allowed him to study the distribution properties of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Groningen Groningen (; gos, Grunn or ) is the capital city and main municipality of Groningen province in the Netherlands. The ''capital of the north'', Groningen is the largest place as well as the economic and cultural centre of the northern part of the country; as of December 2021, it had 235,287 inhabitants, making it the sixth largest city/municipality of the Netherlands and the second largest outside the Randstad. Groningen was established more than 950 years ago and gained city rights in 1245. Due to its relatively isolated location from the then successive Dutch centres of power ( Utrecht, The Hague, Brussels), Groningen was historically reliant on itself and nearby regions. As a Hanseatic city, it was part of the North German trade network, but later it mainly became a regional market centre. At the height of its power in the 15th century, Groningen could be considered an independent city-state and it remained autonomous until the French era. Today Groningen is a universi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Brașov Brașov (, , ; german: Kronstadt; hu, Brassó; la, Corona; Transylvanian Saxon: ''Kruhnen'') is a city in Transylvania, Romania and the administrative centre of Brașov County. According to the latest Romanian census ( 2011), Brașov has a population of 253,200 making it the 7th most populous city in Romania. The metropolitan area is home to 382,896 residents. Brașov is located in the central part of the country, about north of Bucharest and from the Black Sea. It is surrounded by the Southern Carpathians and is part of the historical region of Transylvania. Historically, the city was the center of the Burzenland, once dominated by the Transylvanian Saxons, and a significant commercial hub on the trade roads between Austria (then Archduchy of Austria, within the Habsburg monarchy, and subsequently Austrian Empire) and Turkey (then Ottoman Empire). It is also where the national anthem of Romania was first sung. Names Brassovia, Brassó, Brașov, etc. According to Drago ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Bucharest Bucharest ( , ; ro, București ) is the capital and largest city of Romania, as well as its cultural, industrial, and financial centre. It is located in the southeast of the country, on the banks of the Dâmbovița River, less than north of the Danube River and the Bulgarian border. Bucharest was first mentioned in documents in 1459. The city became the capital of Romania in 1862 and is the centre of Romanian media, culture, and art. Its architecture is a mix of historical (mostly Eclectic, but also Neoclassical and Art Nouveau), interbellum ( Bauhaus, Art Deco and Romanian Revival architecture), socialist era, and modern. In the period between the two World Wars, the city's elegant architecture and the sophistication of its elite earned Bucharest the nickname of 'Paris of the East' ( ro, Parisul Estului) or 'Little Paris' ( ro, Micul Paris). Although buildings and districts in the historic city centre were heavily damaged or destroyed by war, earthquakes, and even Ni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Conjectures In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of ''Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]