Development Of Probability Theory Research Articles

Pareto and probability distributions

AbstractAlthough familiar with the developments in probability theory of his time, Vilfredo Pareto made little use of this tool in his writings, preferring theoretical constructions based on experimentation and observation. This article attempts to reconstruct Pareto’s overall approach to probability by examining his references to the distribution of income, an economic fact that lends itself to probabilistic investigation. The result of this research shows how Pareto alludes to the application of probability to income and social groups, but leaves the task to his followers.JEL classification B31, B4, C1.

Theoretical-probablic approach to the modeling of uncertainty in economic activity from the perspective of the historical path of its development

The article is devoted to the problem of modeling uncertainty in the field of economics and business based on probability theory. Specifically, the purpose of the study was to review the features, potential and limitations of probability theory as a means of quantitative analysis and modeling of uncertainty in economic activity from the point of view of the historical path of its formation and development. As part of this, the key events of the origin and development of probability theory, key aspects of substantiating its logical foundations and building it as a purely mathematical discipline were considered. Particular attention was paid to the issue of distinguishing between stochastic and non-stochastic uncertainty in the plane of practical situations of economic decision-making. As one of the important methodological emphases of the work, the thesis that when the situation of making an economic decision is burdened by non-stochastic uncertainty of one or another origin, it is justified and useful to turn to an expert approach. At the same time, the theory of subjective probabilities, the theory of fuzzy sets, and interval analysis stand out as the system-forming components of the theoretical basis of the latter.

Academician Jonas Kubilius: works dedicated to the history of Lithuanian mathematics

The rise of the Lithuanian mathematical school in the second half of the 20th century is associated with the development of probability theory and its application, and the foundations of that school were insightfully laid by the famous Lithuanian mathematician Jonas Kubilius. However, the academician also had a second vocation – the history of mathematics. At the end of the 20th century, he purposefully researched the mathematical legacy of the poet, bishop A. Baranauskas, recognizing him as the first Lithuanian mathematician researcher of the second half of the 19th century.
 At the beginning of the 21st century, J. Kubilius undertook a detailed implementation of the idea of a work in the history of Lithuanian mathematics. For this purpose, an informal group of specialists was convened, the content of the work was planned, and the research-based book series ``From the History of Lithuanian Mathematics'' was published. The fourth book in this series, Mathematics in Lithuanian Higher Education Institutions in 1921–1944, presents the research of an academic who reveals the situation of mathematics in universities in Kaunas and Vilnius.
 In addition, the memoirs of mathematics history by J. Kubilius, dedicated to mathematicians Z. Žemaitis, G. Žilinskas and V. Statulevičius, should be mentioned.
 The article, at the end of which fragments of the author's memories are presented, is dedicated to the centenary of the birth of Academician J. Kubilius.

Independent events and their complements

ABSTRACT Take a family of independent events. If some of these events, or all of them, are replaced by their complements, then independence still holds. This fact, which is agreed upon by the members of the statistical/probability communities, is tremendously well known, is fairly intuitive and has always been frequently used for easing probability calculations and allowing important developments in probability theory. However, can this result be proven without advanced elements from probability theory, such as independent σ-fields and related results? This paper aims primarily to fill this possible gap. Two elementary proofs that require only introductory probability content are given in detail and compared with the ‘technical’ proof found in books on advanced probability theory. A discussion regarding how these proofs could be included in regular probability courses is also included in the survey.

About teaching students to methods of solving problems by combinator

This article reveals some aspects of the formation of skills to solve combinatorial problems when studying a school course in mathematics. It also considers methods for solving historical combinatorial problems, combinatorial problems and the rule of multiplication, developing skills for solving combinatorial problems, tasks on forming concepts, a tree of options, factorial, applying equations to equations and simplifying expressions, combinatorial problems for studying the concepts of permutations without repetitions, permutations with repetitions, placements without repetitions, placements with repetitions, combinations without repetitions, combinations with repetitions. In mathematics, there are many problems that require elements make available a different set, count the number of all possible combinations of elements formed by a certain rule. Such problems are called combinatorial, and the branch of mathematics involved in solving these problems is called combinatorics. Some combinatorial problems were solved in ancient China, and later in the Roman Empire. However, as an independent branch of mathematics, combinatorics took shape in Europe only in the 18th century. in connection with the development of probability theory. In ancient times, pebbles were often used to facilitate calculations. In this case, special attention was paid to the number of pebbles that could be laid out in the form of a regular figure. So square numbers appeared (1, 4, 16, 25, ...). In everyday life, we often face problems that have not one, but several different solutions. To make the right choice, it is very important not to miss any of them. To do this, iterate through all possible options. Such problems are called combinatorial. It turns out that the multiplication rule for three, four, etc. tests can be explained without going beyond the plane, using a geometric picture (model), which is called the tree of possible options. It, firstly, like any picture, is visual and, secondly, it allows you to take everything into account without missing anything.

Congratulations to A. N. Shiryaev

On December 2, 2015, the University of Angers handed the title and insignia of Doctor Honoris Causa to Russian mathematician and member of the Russian Academy of Sciences, professor Albert N. Shiryaev, for his great contribution to the development of probability theory.

Invariance principles for the law of the iterated logarithm under G-framework

The classical law of the iterated logarithm (LIL for short)as fundamental limit theorems in probability theory play an important role in the development of probability theory and its applications. Strassen (1964) extended LIL to large classes of functional random variables, it is well known as the invariance principle for LIL which provide an extremely powerful tool in probability and statistical inference. But recently many phenomena show that the linearity of probability is a limit for applications, for example in finance, statistics. As while a nonlinear expectation--- G-expectation has attracted extensive attentions of mathematicians and economists, more and more people began to study the nature of the G-expectation space. A natural question is: Can the classical invariance principle for LIL be generalized under G-expectation space? This paper gives a positive answer. We present the invariance principle of G-Brownian motion for the law of the iterated logarithm under G-expectation.

What is probability theory?

This issue of Resonance features Joseph Leonard Doob, who played a critical role in the development of probability theory in the world from 1935 onwards. The goal of the present article is to explain to the readers of Resonance what probability theory is all about.

Joseph l doob and development of probability theory

Joseph l doob and development of probability theory

Some moment inequalities for fuzzy martingales and their applications

Abstract Martingales are a class of stochastic processes which has had profound influence on the development of probability theory and stochastic processes. Some recent developments are related to mathematical finance. In the real world, some information about these phenomena might be imprecise and represented in the form of vague quantities. In these situations, we need to generalize classical methods to vague environment. Thus, fuzzy martingales have been extended as a vague perception of real-valued martingales. In this paper, some moment inequalities are presented for fuzzy martingales. Several convergence theorems are established based on these inequalities. As an application of convergence theorems, a weak law of large numbers for fuzzy martingales is stated. Furthermore, a few examples are devoted to clarify the main results.

A History of Evidence in Medical Decisions

Bayesian inference in medical decision making is a concept that has a long history with 3 essential developments: 1) the recognition of the need for data (demonstrable scientific evidence), 2) the development of probability, and 3) the development of inverse probability. Beginning with the demonstrative evidence of the physician's sign, continuing through the development of probability theory based on considerations of games of chance, and ending with the work of Jakob Bernoulli, Laplace, and others, we will examine how Bayesian inference developed.

Lectures for chemists on statistics. I. Belief, probability, frequency, and statistics: decision making in a floating world

A brief account on the development of probability theory and statistics is given. These subjects may help the chemist to interpret the information content of his experimental data. Their source and validity, however, is often unclear. A series of papers, of which this is the first, intends to present a clearer view by illustrating the origin of the data-analysis tools. The major figures in this field are mentioned, their achievements presented and the mutual links outlined. The intention is the provision of a common basis for the discussion of statistical aspects relevant to metrology in chemistry. The chemist should see probability theory and statistics as tools to assist in the daily practice of data interpretation to arrive at an objective conclusion.

Probability in 1919/20: the von Mises-Pólya-Controversy

The correspondence between Richard von Mises and George Polya of 1919/20 contains reflections on two well-known articles by von Mises on the foundations of probability in the Mathematische Zeitschrift of 1919, and one paper from the Physikalische Zeitschrift of 1918. The topics touched on in the correspondence are: the proof of the central limit theorem of probability theory, von Mises' notion of randomness, and a statistical criterion for integer-valuedness of physical data. The investigation will hint at both the fruitfulness and the limits of several of von Mises' notions such as ``collective'', ``distribution'' and ``complex adjuncts'' (characteristic functions) for further developments in probability theory and in ``directional statistics''. By pointing to the selectiveness of Polya's criticism, the historical analysis shows the differing expectations of the two men with respect to the further development of the theory of probability and its applications. The paper thus gives a glimpse of the provisional state of the theory around 1920, before others such as P. Levy (1886–1971) and A. N. Kolmogorov (1903–1987) stepped in and created a new paradigm for probability theory.

Some remarks on the joint observable in D-posets

The notion of the joint observable contributed, besides other things, to the development of probability theory on structures which are more general than σ-algebras. We present some remarks on the joint observable in D-posets which arising in the definition of operations with observables.

On the Creative Contribution of V. S. Korolyuk to the Development of Probability Theory

We present a brief survey of the main results obtained by V. S. Korolyuk in probability theory and mathematical statistics.

Some Remarks on the Article “On the Creative Contribution of V. S. Korolyuk to the Development of Probability Theory”

1. Asymptotic Expansions for the Goodness-of-Fit Tests of Kolmogorov and Smirnov. Asymptotic expansions for the goodness-of-fit tests of Kolmogorov and Smirnov were first constructed without taking into account the influence of boundary layers [4 – 7]. Later, on the basis of the exact formulas for the Kolmogorov and Smirnov distributions, Jiang Li-shang, a Chinese mathematician, constructed the first three terms of the asymptotic expansion, and it became clear that it is necessary to take into account additional terms of the asymptotics related to the effect of a jump over the boundary. It is now difficult to explain why the works of Vishik and Lyusternik drew my attention because, in their works, the degeneration of boundary conditions meant only the loss of additional values of higher derivatives. However, the key idea of necessity of constructing boundary layers was undeniable. The main problem was to decompose the original operator P on functions in the neighborhood of the boundary of the domain. To solve this problem, I used the results obtained by Krein in the theory of integral equations of the convolution type on a semiaxis. The factor-

Discussion

This is a discussion of the Forum Lectures by Evarist Ginéon the subject of Empirical Processes and Applications presented at the European Meeting of Statisticians held in Bath, England, September 13-18, 1992. The discussion includes short sketches of developments in probability theory (Gaussian process theory, weak convergence theory, and probability in Banach spaces), empirical process theory, and applications thereof in statistics. I comment briefly on the formulation of central limit theorems for empirical processes in terms of the presence or absence of a Gaussian hypothesis, expand on Professor Giné's discussion of the bootstrap, and briefly explain my recent results for exchangeably weighted bootstraps obtained jointly with Jens Praestgaard. The discussion closes with some problems.

Rejoinder

This is a rejoinder of the Forum Lectures by Evarist Ginéon the subject of Empirical Processes and Applications presented at the European Meeting of Statisticians held in Bath, England, September 13-18, 1992. The discussion includes short sketches of developments in probability theory (Gaussian process theory, weak convergence theory, and probability in Banach spaces), empirical process theory, and applications thereof in statistics. I comment briefly on the formulation of central limit theorems for empirical processes in terms of the presence or absence of a Gaussian hypothesis, expand on Professor Giné's discussion of the bootstrap, and briefly explain my recent results for exchangeably weighted bootstraps obtained jointly with Jens Praestgaard. The discussion closes with some problems.

SOME ASPECTS OF INFERENCE FOR MULTIVARIATE INFINITELY DIVISIBLE DISTRIBUTIONS

Abstract : Measurement of dependence in the infinitely divisible class of multivariate distributions, based on developments in probability theory for that class, is discussed. It has been shown that pairwise independence is equivalent to mutual independence in the infinitely divisible class. When the infinitely divisible variables contain no normal component (in particular, when they are discrete), the cumulant of order (2,2) can be used as a measure of pairwise dependence; when a normal component is present, the appropriate measure of pairwise dependence also involves the covariance. Results for testing independence of infinitely divisible random variables are discussed. A method of testing normality against infinitely divisible alternatives is given. (Author)

Banach space-valued random variables and tensor products of Banach spaces

Since the publication in 1953 of Mourier’s paper [9] on random variables with values in a Banach space, a large number of papers concerned with the development of probability theory in Banach spaces have appeared. These papers have, in the main, been devoted to (i) measure-theoretic problems, (ii) limit theorems, (iii) martingale theory, and (iv) the theory of random equations. We refer to Bharucha-Reid [l], Driml and HanH [4], and Grenander [6] for references. Many studies in abstract probability theory and its applications lead to the spaces L,(Q, @, p, X) = L,(Q, X) o random variables defined on a probaf bility space (Q, ed, p) with values in a Bancah spaceX In this paper we utilize tensor product methods, as applied to the L,(Q, X) spaces, to consider a number of problems. In Section 2 we discuss (i) strong random variables, (ii) the spaces L&2, X) as tensor product Banach spaces L,(Q) B X, and (iii) the connections between the norms on the spaces L&2, X) and L&Q, X) and the y and h norms in the sense of tensor product crossnorms (cf. Schatten [14]). In Section 3 we utilize tensor product methods to prove the existence of the conditional expectation for strong random variables. Sections 4 and 5 are devoted to tensor product Hilbert spaces, and in Section 6 we consider a problem concerning the eigenvalues of a random Hermitian matrix. In Section 4 we show that every pair of elements of a tensor product Hilbert space H @ 8, where H and sj are Hilbert spaces,