Problem In Probability Theory Research Articles

Euler on Statistics

This paper gives a short review of Euler’s work in the statistical sciences. During Euler’s lifetime there was a strong interest in the practical use of mathematical modelling and this increasingly encompassed the calculus of probabilities. In the middle of his life, Euler’s interest was piqued by questions involving games of chance. He also occasionally solved problems in probability theory. At the request of king Frederic II of Prussia, for example, he analysed designs for two state run lotteries. The natural applications of probability and statistics involved such questions of public policies as the prediction of population growth, or the establishment of life tables in order to predict the expenses of insurance schemes for widows and orphans. A few of Eulers texts in these areas are discussed in the paper.

Explicit representation of roots on p-adic solenoids and non-uniqueness of embeddability into rational one-parameter subgroups

This note generalizes known results concerning the existence of roots and embedding one-parameter subgroups on p-adic solenoids. An explicit representation of the roots leads to the construction of two distinct rational embedding one-parameter subgroups. The results contribute to enlighten the group structure of solenoids and to point out difficulties arising in the context of the embedding problem in probability theory. As a consequence, the uniqueness of embedding of infinitely divisible probability measures on p-adic solenoids is solved under a certain natural condition.

On a problem in probability theory

For continuous random variables, we study a problem similar to that considered earlier by one of the authors for discrete random variables. Let numbers $$N > 0, E > 0, 0 \leqslant \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _s $$ be given. Consider a random vector x = (x1, …, xs), uniformly distributed on the set $$x_j \geqslant 0, j = 1, \ldots ,s; \sum\limits_{j = 1}^s {x_j = N} , \sum\limits_{j = 1}^s {\lambda _j x_j \leqslant E} .$$ We study the weak limit of x as s → ∞.

A Certain Class of Laplace Transforms with Applications to Reaction and Reaction-Diffusion Equations

A class of Laplace transforms is examined to show that particular cases of this class are associated with production-destruction and reaction-diffusion problems in physics, study of differences of independently distributed random variables and the concept of Laplacianness in statistics, α-Laplace and Mittag-Leffler stochastic processes, the concepts of infinite divisibility and geometric infinite divisibility problems in probability theory and certain fractional integrals and fractional derivatives. A number of applications are pointed out with special reference to solutions of fractional reaction and reaction-diffusion equations and their generalizations.

Bertrand's chord, Buffon's needle, and the concept of randomness

Two old problems in probability theory involving the concept of randomness are considered. Data obtained with one of them—Bertrand's chord problem—demonstrate the equivocality of this term in the absence of a definition or explication of assumptions underlying its use. They also support two propositions about probabilistic thinking: (1) upon obtaining an answer to a question of probability, people tend to see it as the answer, overlooking tacit assumptions on which it may be based, and tend not to consider the possibility of other assumptions from which different answers would follow; (2) perceptual balance—moderate uniformity of density of distribution—is a determinant of the degree to which a two-dimensional pattern is perceived as random.

The Difference Between Selection and Drift: A Reply to Millstein

Millstein [Bio. Philos. 17 (2002) 33] correctly identies a serious problem with the view that natural selection and random drift are not conceptually distinct. She offers a solution to this problem purely in terms of differences between the processes of selection and drift. I show that this solution does not work, that it leaves the vast majority of real biological cases uncategorized. However, I do think there is a solution to the problem she raises, and I offer it here. My solution depends on solving the biological analogue of the reference class problem in probability theory and on the reality of individual fitnesses.

A new method of solution for the occupancy problem and its application to operon size prediction

The problem of estimating the expected number of transcription units containing a specific number of genes arises in the context of operon size prediction in prokaryotic genomes, where operons are defined to be transcription units containing two or more genes. It turns out that this problem is identical mathematically to the balls in urns occupancy problem in probability theory. In that problem, a fixed number of indistinguishable balls are randomly placed in a known number of distinguishable urns, subject to the restriction that no urns may remain empty, and an estimate is desired for the expected number of urns containing a specific number of balls. In this paper we present a new simple technique for solving the occupancy problem when empty urns are allowed and extend it to the case when each urn must contain the same non-zero minimum number of balls. Treating transcription units as equivalent to urns, and genes as equivalent to balls, we then use that result to solve the problem of estimating the expected number of transcription units that contain a specific number of genes, and then apply that result to predicting the expected number of transcription units present in an entire genome. Since these predictions can be made for any completely sequenced and annotated prokaryotic genome, they provide a starting point for the comparison of regulatory complexity across such genomes.

Quantum Statistics Methods from the Viewpoint of Probability Theory. I

We show that one can assign a self-adjoint operator in a Hilbert space (a symmetric matrix in the finite-dimensional case) to standard problems in probability theory and introduce the notions of entropy, temperature, and statistical ensemble. A series of general identities for these variables result, in particular, in the Bardeen--Cooper formulas for superconductivity. A rigorous proof of these asymptotics and their applications will be given in the second part of the paper.

Econophysics and Quantum Statistics

In this paper, we associate notions of entropy, temperature, free energy, and Hamiltonian occuring in problems in probability theory with a system of identical objects so that modern methods of quantum statistics can be applied to problems in mathematical finance.

Some applied problems of stability of stochastic models

We consider several applied problems of probability theory. These problems are related to biophysics, mathematical economics, and automated control theory, and are considered in the framework of the theory of stability of stochastic models.

A framework for linear information inequalities

We present a framework for information inequalities, namely, inequalities involving only Shannon's information measures, for discrete random variables. A region in IR(2/sup n/-1), denoted by /spl Gamma/*, is identified to be the origin of all information inequalities involving n random variables in the sense that all such inequalities are partial characterizations of /spl Gamma/*. A product from this framework is a simple calculus for verifying all unconstrained and constrained linear information identities and inequalities which can be proved by conventional techniques. These include all information identities and inequalities of such types in the literature. As a consequence of this work, most identities and inequalities involving a definite number of random variables can now be verified by a software called ITIP which is available on the World Wide Web. Our work suggests the possibility of the existence of information inequalities which cannot be proved by conventional techniques. We also point out the relation between /spl Gamma/* and some important problems in probability theory and information theory.

The Research of Wim Vervaat

This essay gives an outline of some of Wim Vervaat's innovative research ideas, especially those involving sup measures and capacities. These ideas were developed at his initiative in order to find satisfactory solutions to some limit problems in probability theory. The essay discusses the motivating questions and the role of sup measures and capacities in their solutions.

REFERENCE CLASSES AND MULTIPLE INHERITANCES

The reference class problem in probability theory and the multiple inheritances (extensions) problem in non-monotonic logics are special cases of conflicting beliefs. One popular solution accepted in the two domains is the specificity principle. By analyzing an example, several factors beyond specificity are found to be relevant to the priority of a reference class. A new approach, Non-Axiomatic Reasoning System (NARS), is discussed, where these factors are all taken into account. It is argued that the solution provided by NARS is better than the solutions provided by probability theory and non-monotonic logics.

Asymptotic Problems in Probability Theory: Wiener Functionals and Fractionals.

Asymptotic Problems in Probability Theory: Wiener Functionals and Fractionals.

The Development of the Mathematical Theory of Extremes in the Past Half Century

This paper is a survey of development of extreme value theory during the last half century from a mathematician’s point of view. The paper considers both general results for classical models (schemes of maximum of independent identically distributed random variables under linear normalization) and extentions of the classical models and the derivations from the classical assumptions also. Different approaches to the characterization theorems for limiting distributions are discribed. The results on the estimation of the rate of convergence in limiting theorems and random sample sizes are given. The paper gives a survey of different approaches to the extreme value theory based on other problems of probability theory.

Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals.

Limits for stochastic models: stochastic and quantum mechanical scattering on hyperbolic spaces, E.A. Carlen and K.D. Elsworthy intermittency and phase transitions from some particle systems in random media, R. Carmona and S.A. Molchanov predator-prey systems, R. Durrett homogenization of reflecting barrier Brownian motions, H. Osada nonlinear diffusion limit for a system with nearest neighbour interactions-II, S.R.S. Varadhan. Part 2 Analysis on fractals: random walks, electrical resistance, and nested fractals, M.R. Barlow Hausdorff dimension of nonlinear Cantor sets, K. Handa self-avoiding process on the Sierpinski gasket, K. Hatton harmonic metric and Dirichlet form on the Sierpinski gasket, J. Kigami construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket, T. Kumagai Brownian motion penetrating the Sierpinski gasket, T. Lindstrom the eigenvalue problem for the Laplacian on the Sierpinski gasket, T. Shima.

How to Look at Objects in a 5-Dimensional Shape Space. I: Looking at Distributions

This paper is a survey of development of extreme value theory during the last half century from a mathematician’s point of view. The paper considers both general results for classical models (schemes of maximum of independent identically distributed random variables under linear normalization) and extentions of the classical models and the derivations from the classical assumptions also. Different approaches to the characterization theorems for limiting distributions are discribed. The results on the estimation of the rate of convergence in limiting theorems and random sample sizes are given. The paper gives a survey of different approaches to the extreme value theory based on other problems of probability theory.

On a Refinement of the Central Limit Theorem for Sums of Independent Random Indicators

Previous article Next article On a Refinement of the Central Limit Theorem for Sums of Independent Random IndicatorsV. G. MikhailovV. G. Mikhailovhttps://doi.org/10.1137/1138044PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractExplicit and rather tight upper bounds for the distance (in the uniform metric) between the distribution function of a sum of independent random indicators and its asymptotic expansion are obtained.[1] V. V. Petrov, Limit Theorems for Sums of Independent Random Variables, Nauka, Moscow, 1987, (In Russian.) Google Scholar[2] V. M. Zolotarev, Real refinements of limit theorems in probability theory, Trudy Mat. Inst. Steklov., 182 (1988), 24–47, (In Russian.) 89k:60038 Google Scholar[3] V. A. Vatutin and , V. G. Mikhailov, Limit theorems for the number of empty cells in equiprobable scheme for group allocation of particles, Theory Probab. Appl., 27 (1982), 734–743 10.1137/1127084 0536.60017 LinkGoogle Scholar[4] Valentin F. Kolchin, , Boris A. Sevastyanov and , Vladimir P. Chistyakov, Random allocations, V. H. Winston & Sons, Washington, D.C., 1978xi+262 57:10758b 0376.60003 Google Scholar[5] V. A. Ivanov, , C. I. Ivchenko and , Yu. I. Medvedev, Discrete problems in probability theory, Soviet Math., 31 (1985), 2– 0579.62001 Google Scholar[6] Paul Deheuvels, , Madan L. Puri and , Stefan S. Ralescu, Asymptotic expansions for sums of nonidentically distributed Bernoulli random variables, J. Multivariate Anal., 28 (1989), 282–303 10.1016/0047-259X(89)90111-5 90k:62039 0676.60024 CrossrefGoogle Scholar[7] William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons Inc., New York, 1966xviii+636 35:1048 0138.10207 Google Scholar[8] S. N. Bernstein, Collected Works, Vol. 4, Nauka, Moscow, 1964, (In Russian.) Google Scholar[9] L. N. Bolshev and , N. V. Smirnov, Tables of Mathematical Statistics, Nauka, Moscow, 1983, (In Russian.) 0529.62099 Google ScholarKeywordsrandom indicatorsnonhomogenuous Bernoulli schemeasymptotic expansioncloseness of approximation Previous article Next article FiguresRelatedReferencesCited byDetails The Chebyshev--Edgeworth Correction in the Central Limit Theorem for Integer-Valued Independent SummandsS. G. Bobkov and V. V. Ulyanov3 February 2022 | Theory of Probability & Its Applications, Vol. 66, No. 4AbstractPDF (426 KB)On the Computation and Approximation of Outage Probability in Satellite Networks With Smart Gateway DiversityIEEE Transactions on Aerospace and Electronic Systems, Vol. 57, No. 1 Cross Ref Поправка Чебышeва-Эджворта в центральной предельной теореме для целочисленных независимых слагаемых22 October 2021 | Теория вероятностей и ее применения, Vol. 66, No. 4 Cross Ref A simple and fast method for computing the Poisson binomial distribution functionComputational Statistics & Data Analysis, Vol. 122 Cross Ref Inferring monopartite projections of bipartite networks: an entropy-based approach17 May 2017 | New Journal of Physics, Vol. 19, No. 5 Cross Ref Sparse covers for sums of indicators2 November 2014 | Probability Theory and Related Fields, Vol. 162, No. 3-4 Cross Ref Learning Poisson Binomial Distributions11 February 2015 | Algorithmica, Vol. 72, No. 1 Cross Ref Asymptotic identifiability of nonparametric item response modelsPsychometrika, Vol. 66, No. 4 Cross Ref Binomial Approximation to the Poisson Binomial Distribution: The Krawtchouk Expansion25 July 2006 | Theory of Probability & Its Applications, Vol. 45, No. 2AbstractPDF (184 KB)A Refinement of the Central Limit Theorem for Sums of Independent Random Indicators12 July 2006 | Theory of Probability & Its Applications, Vol. 40, No. 4AbstractPDF (442 KB) Volume 38, Issue 3| 1994Theory of Probability & Its Applications History Submitted:04 September 1990Published online:17 July 2006 InformationCopyright © 1993 Society for Industrial and Applied MathematicsKeywordsrandom indicatorsnonhomogenuous Bernoulli schemeasymptotic expansioncloseness of approximationPDF Download Article & Publication DataArticle DOI:10.1137/1138044Article page range:pp. 479-489ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Problems of Probability Theory

Problems of Probability Theory

On some aspects of survival under production uncertainty

In recent years there has emerged a considerable volume of literature which analyzes the possibility of a certain performance index staying above a minimum level over time for an agent who operates in a given stochastic dynamic environment. Thus, one may be interested in the potential ability of an agent to meet a minimum positive "subsistence" consumption level over time or of firms being able to pay out a minimum level of dividend to shareholders or to meet a minimal level of debt service in order to avoid bankruptcy. For a renewable resource which is exploited over time at a rate determined by the market or by the optimal policy of its owner, one can examine the circumstances under which the resource does not become extinct over time. The literature on exhaustible resources has focused on the difficulties of maintaining a positive steady level of consumption in an economy which relies on an exhaustible resource as an essential input in production [see Solow (1974), Cass and Mitra (1991)]. In a model of renewable resource with stochastic and concave production function, where the resource is depleted every time period according to market equilibrium, Mirman and Spulber (1984) derived results on chances of survival for the resource. Majumdar and Radner (1991,1992) consider a dynamic model ?f consumption and investment with production uncertainty, where the agent is required to meet a certain strictly positive consumption level every time period in order to survive. They consider cases where the production function is of the canonical neoclassical type and also the case where it is linear. This framework is close to the classical gambler's ruin problem in probability theory (see Feller (1957), Billingsley (1979)). In a similar framework, Ray (1984) analyzed the survival problem of economic agents who had the option of borrowing and lending over time.