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Journal articles on the topic 'Geometric theory probability'

Author: Grafiati

Published: 28 May 2022

Last updated: 31 July 2025

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1

Holik, Federico, Cesar Massri, and A. Plastino. "Geometric probability theory and Jaynes’s methodology." International Journal of Geometric Methods in Modern Physics 13, no. 03 (2016): 1650025. http://dx.doi.org/10.1142/s0219887816500250.

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We provide a generalization of the approach to geometric probability advanced by the great mathematician Gian Carlo Rota, in order to apply it to generalized probabilistic physical theories. In particular, we use this generalization to provide an improvement of the Jaynes’ MaxEnt method. The improvement consists in providing a framework for the introduction of symmetry constraints. This allows us to include group theory within MaxEnt. Some examples are provided.

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2

Vakhrameev, S. A., and A. V. Sarychev. "Geometric control theory." Journal of Soviet Mathematics 40, no. 3 (1988): 384–447. http://dx.doi.org/10.1007/bf01092893.

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3

Hykšová, Magdalena, Anna Kalousová, and †Ivan Saxl. "EARLY HISTORY OF GEOMETRIC PROBABILITY AND STEREOLOGY." Image Analysis & Stereology 31, no. 1 (2012): 1. http://dx.doi.org/10.5566/ias.v31.p1-16.

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The paper provides an account of the history of geometric probability and stereology from the time of Newton to the early 20th century. It depicts the development of two parallel ways: on one hand, the theory of geometric probability was formed with minor attention paid to other applications than those concerning spatial chance games. On the other hand, practical rules of the estimation of area or volume fraction and other characteristics, easily deducible from geometric probability theory, were proposed without the knowledge of this branch. A special attention is paid to the paper of J.-É. Ba

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4

Chen, W. W. L. "Davenport's theorem in geometric discrepancy theory." International Journal of Number Theory 11, no. 05 (2015): 1437–49. http://dx.doi.org/10.1142/s1793042115400047.

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Davenport's theorem was established nearly a lifetime ago, but there have been some very interesting recent developments. The various proofs over the years bring in different ideas from number theory, probability theory, analysis and group theory. In this short survey, we shall not present complete proofs, but will describe instead some of these underlying ideas.

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5

Tyurin, Yu N. "Multivariate Statistical Analysis: The Geometric Theory." Theory of Probability & Its Applications 55, no. 1 (2011): 91–109. http://dx.doi.org/10.1137/s0040585x97984620.

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6

Šikić, Hrvoje, Renming Song, and Zoran Vondraček. "Potential Theory of Geometric Stable Processes." Probability Theory and Related Fields 135, no. 4 (2005): 547–75. http://dx.doi.org/10.1007/s00440-005-0470-3.

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7

Gates, D. J. "Asymptotics of two integrals from optimization theory and geometric probability." Advances in Applied Probability 17, no. 4 (1985): 908–10. http://dx.doi.org/10.2307/1427094.

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8

Gates, D. J. "Asymptotics of two integrals from optimization theory and geometric probability." Advances in Applied Probability 17, no. 04 (1985): 908–10. http://dx.doi.org/10.1017/s0001867800015470.

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9

Vakhrameev, S. A. "Morse theory and Lyusternik-Shnirelman theory in geometric control theory." Journal of Mathematical Sciences 71, no. 3 (1994): 2434–85. http://dx.doi.org/10.1007/bf02111558.

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10

Gbolagade, A. M., and I. T. Awolere. "GENERALIZED DISTRIBUTION FOR BI-UNIVALENT FUNCTIONS DEFINED BY ERROR AND POISSON DISTRIBUTION VIA BELL NUMBER." COAST JOURNAL OF THE SCHOOL OF SCIENCE 6, no. 2 (2024): 1120–28. https://doi.org/10.61281/coastjss.v6i2.12.

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In the field of geometric function theory, generalized distributions have revealed novel insights and applications, particularly in understanding the behaviour of various complex functions. This paper focuses on estimating bounds for bi-univalent functions within probability distribution series defined by error and Poisson distributions, particularly in relation to the Bell numbers. These distributions are utilized to establish coefficient bounds, which hold significance for both the structural properties of bi-univalent functions and their applications in probability theory. By extending thes

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11

Cheng, Qiang, Can Wu, Peihua Gu, Wenfen Chang, and Dongsheng Xuan. "An Analysis Methodology for Stochastic Characteristic of Volumetric Error in Multiaxis CNC Machine Tool." Mathematical Problems in Engineering 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/863283.

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Traditional approaches about error modeling and analysis of machine tool few consider the probability characteristics of the geometric error and volumetric error systematically. However, the individual geometric error measured at different points is variational and stochastic, and therefore the resultant volumetric error is aslo stochastic and uncertain. In order to address the stochastic characteristic of the volumetric error for multiaxis machine tool, a new probability analysis mathematical model of volumetric error is proposed in this paper. According to multibody system theory, a mean val

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12

Agrachev, A. A. "Quadratic mappings in geometric control theory." Journal of Soviet Mathematics 51, no. 6 (1990): 2667–734. http://dx.doi.org/10.1007/bf01095430.

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13

CASTRO, CARLOS, and JORGE MAHECHA. "FRACTAL SUPERSYMMETRIC QM, GEOMETRIC PROBABILITY AND THE RIEMANN HYPOTHESIS." International Journal of Geometric Methods in Modern Physics 01, no. 06 (2004): 751–93. http://dx.doi.org/10.1142/s0219887804000393.

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The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn=1/2+iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss–Jacobi theta series, allows us to provide the proper framework to construct the well-defined algorithm to compute the density of zeros in the critical line, which would complement the existing formulas in the literature for the density of zeros in the critical strip. Geometric probability theory furnishes the answer to the difficult question whether the probability that the RH is true

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14

Bhadula, Rakesh Chandra. "Investigation of Representation Theory for Algebraic and Geometric Applications." Mathematical Statistician and Engineering Applications 70, no. 1 (2021): 580–86. http://dx.doi.org/10.17762/msea.v70i1.2511.

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Understanding the algebraic and geometric structures that form in diverse mathematical areas depends heavily on the study of representation theory. The significance and uses of representation theory in both algebra and geometry are briefly discussed in this abstract.The primary goal of representation theory is to understand how linear transformations on vector spaces can represent abstract algebraic objects like groups, rings, and algebras. Representation theory offers a strong framework to analyse and interact with these structures using the methods and tools of linear algebra by linking alge

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15

Krushkal, Samuel L. "Teichmüller space theory and classical problems of geometric function theory." Journal of Mathematical Sciences 258, no. 3 (2021): 276–89. http://dx.doi.org/10.1007/s10958-021-05546-5.

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16

Tsar’kov, I. G. "Some Applications of Geometric Theory of Approximations." Journal of Mathematical Sciences 245, no. 1 (2020): 64–82. http://dx.doi.org/10.1007/s10958-020-04677-5.

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17

Blazhennova-Mikulich, L. Ju. "On some problems in geometric game theory." Journal of Mathematical Sciences 147, no. 2 (2007): 6639–43. http://dx.doi.org/10.1007/s10958-007-0500-z.

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18

Chen, Yanhua, Ubaid ur Rehman, and Tahir Mahmood. "Bipolar Fuzzy Multi-Criteria Decision-Making Technique Based on Probability Aggregation Operators for Selection of Optimal Artificial Intelligence Framework." Symmetry 15, no. 11 (2023): 2045. http://dx.doi.org/10.3390/sym15112045.

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Artificial intelligence (AI) frameworks are essential for development since they offer pre-built tools and libraries that speed up and simplify the production of AI models, leveraging symmetry to save time and effort. They guarantee effective computing by modifying code for particular hardware, facilitating quicker testing and deployment. The identification of a suitable and optimal AI framework for development is a multi-criteria decision-making (MCDM) dilemma, where the considered AI frameworks for development are evaluated by considering various criteria and these criteria may have dual asp

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19

Xia, Aihua, and J. E. Yukich. "Normal approximation for statistics of Gibbsian input in geometric probability." Advances in Applied Probability 47, no. 4 (2015): 934–72. http://dx.doi.org/10.1239/aap/1449859795.

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This paper concerns the asymptotic behavior of a random variable Wλ resulting from the summation of the functionals of a Gibbsian spatial point process over windows Qλ ↑ ℝd. We establish conditions ensuring that Wλ has volume order fluctuations, i.e. they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Stein's method to deduce rates of a normal approximation for Wλ as λ → ∞. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input.

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20

Xia, Aihua, and J. E. Yukich. "Normal approximation for statistics of Gibbsian input in geometric probability." Advances in Applied Probability 47, no. 04 (2015): 934–72. http://dx.doi.org/10.1017/s0001867800048965.

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This paper concerns the asymptotic behavior of a random variableWλresulting from the summation of the functionals of a Gibbsian spatial point process over windowsQλ↑ℝd. We establish conditions ensuring thatWλhas volume order fluctuations, i.e. they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Stein's method to deduce rates of a normal approximation forWλas λ → ∞. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input. We also

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21

Vaško, Milan, and Milan Sága. "Application of Fuzzy Structural Analysis for Damage Prediction Considering Uncertain S/N Curve." Applied Mechanics and Materials 420 (September 2013): 21–29. http://dx.doi.org/10.4028/www.scientific.net/amm.420.21.

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The paper presents chosen traditional (based on probability theory) and non-traditional (based on possibility theory) computational tools for analysis of the material, geometric or loading uncertainties in mechanical structures. Uncertainties are introduced as bounded possible values intervals or as fuzzy sets, assuming possibility theory and as random parameters in the case of the probability theory. The main goal was to propose numerical algorithms for fuzzy analysis of stochastic oscillated FE model (truss structure) and to predict fuzzy fatigue damage considering fuzzy S/N curve.

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22

Hake, Kathleen. "Knotting probability of equilateral hexagons." Journal of Knot Theory and Its Ramifications 28, no. 07 (2019): 1950040. http://dx.doi.org/10.1142/s0218216519500408.

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For a positive integer [Formula: see text], the collection of [Formula: see text]-sided polygons embedded in [Formula: see text]-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded [Formula: see text]-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components correspond to equilateral knot types. When [Formula: see text], the space of equilateral knots is connected. Therefore, we examine the space of equilateral hexagons. Using techniques from symplectic geometry, we can param

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23

Jansons, Kalvis M., and Christopher G. Phillips. "On the application of geometric probability theory to polymer networks and suspensions, I." Journal of Colloid and Interface Science 137, no. 1 (1990): 75–91. http://dx.doi.org/10.1016/0021-9797(90)90044-o.

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24

Cipra, Tomáš. "Autoregressive processes in optimization." Journal of Applied Probability 25, no. 2 (1988): 302–12. http://dx.doi.org/10.2307/3214438.

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Vector autoregressive processes of the first order are considered which are non-negative and optimize a linear objective function. These processes may be used in stochastic linear programming with a dynamic structure. By using Tweedie's results from the theory of Markov chains, conditions for geometric rates of convergence to stationarity (i.e. so-called geometric ergodicity) and for existence and geometric convergence of moments of these processes are obtained.

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25

Giunashvili, Z. "Methods of geometric control theory for quantum computations." Journal of Mathematical Sciences 141, no. 2 (2007): 1113–33. http://dx.doi.org/10.1007/s10958-007-0039-z.

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26

Flimmel, Daniela, Zbyněk Pawlas, and J. E. Yukich. "Limit theory for unbiased and consistent estimators of statistics of random tessellations." Journal of Applied Probability 57, no. 2 (2020): 679–702. http://dx.doi.org/10.1017/jpr.2020.4.

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AbstractWe observe a realization of a stationary weighted Voronoi tessellation of the d-dimensional Euclidean space within a bounded observation window. Given a geometric characteristic of the typical cell, we use the minus-sampling technique to construct an unbiased estimator of the average value of this geometric characteristic. Under mild conditions on the weights of the cells, we establish variance asymptotics and the asymptotic normality of the unbiased estimator as the observation window tends to the whole space. Moreover, weak consistency is shown for this estimator.

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27

Gates, John. "Some Dual Problems of Geometric Probability in the Plane." Combinatorics, Probability and Computing 2, no. 1 (1993): 11–23. http://dx.doi.org/10.1017/s0963548300000456.

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A definition is adopted for convexity of a set of directed lines in the plane. Following this, the duals of a number of standard problems of geometric probability are formulated. Problems considered in detail are the duals of Sylvester's problem, chord length distributions and Ambartzumian's combinatorial geometry. The paper suggests some questions for further work.

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28

Pineda-León, Ernesto, José Manuel Rosales-Juárez, Dante Tolentino, and Orlando Susarrey-Huerta. "Probability of Exceeding Damage States in Plates Using BEM." Advances in Materials Science and Engineering 2019 (November 21, 2019): 1–9. http://dx.doi.org/10.1155/2019/1354065.

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An approach to obtain fragility curves taking into account the formulation for shear deformable plate theory with combined geometric and material nonlinearities and the boundary element method is proposed. It is assumed that the material undergoes large deflection with small strains. The von Mises yield criterion is used to evaluate the plastic zone and is supposed to have elastic-perfectly plastic material behaviour. An initial stress formulation is used to formulate the boundary integral equations. The domain integrals are evaluated using a cell discretization technique. A total incremental

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29

Bera, Asish, Debotosh Bhattacharjee, and Mita Nasipuri. "Fusion-Based Hand Geometry Recognition Using Dempster–Shafer Theory." International Journal of Pattern Recognition and Artificial Intelligence 29, no. 05 (2015): 1556005. http://dx.doi.org/10.1142/s0218001415560054.

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This paper presents a new technique for user identification and recognition based on the fusion of hand geometric features of both hands without any pose restrictions. All the features are extracted from normalized left and right hand images. Fusion is applied at feature and also at decision level. Two probability-based algorithms are proposed for classification. The first algorithm computes the maximum probability for nearest three neighbors. The second algorithm determines the maximum probability of the number of matched features with respect to a thresholding on distances. Based on these tw

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30

Loc Hung, Tran, and Phan Tri Kien. "The necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution." Science and Technology Development Journal 22, no. 1 (2019): 143–46. http://dx.doi.org/10.32508/stdj.v22i1.1049.

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The geometric sums have been arisen from the necessity to resolve practical problems in ruin prob- ability, risk processes, queueing theory and reliability models, etc. Up to the present, the results related to geometric sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned domains of geometric attraction are still limitative. The main purpose of this article is to introduce concepts on the domain of geometric attraction of standard Laplace distribution. Using method of character

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31

Gallo, Sandro, and Pablo M. Rodriguez. "Frog models on trees through renewal theory." Journal of Applied Probability 55, no. 3 (2018): 887–99. http://dx.doi.org/10.1017/jpr.2018.56.

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Abstract We study a class of growing systems of random walks on regular trees, known as frog models with geometric lifetime in the literature. With the help of results from renewal theory, we derive new bounds for their critical parameters. Our approach also improves the existing bounds for the critical parameter of a percolation model on trees known as cone percolation.

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32

Velazquez, L. "Geometric aspects in extended approach of equilibrium classical fluctuation theory." Journal of Statistical Mechanics: Theory and Experiment 2011, no. 11 (2011): P11007. http://dx.doi.org/10.1088/1742-5468/2011/11/p11007.

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33

Bauer, Christina, and Rolf Schneider. "Extremal problems for geometric probabilities involving convex bodies." Advances in Applied Probability 27, no. 1 (1995): 20–34. http://dx.doi.org/10.2307/1428092.

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The theory of geometric probabilities is concerned with randomly generated geometric objects. The aim is to compute probabilities of certain geometric events or distributions of random variables defined in a geometric way. Very often the computation even of simple expectations is too difficult, and one has to be satisfied with establishing estimates and, if possible, sharp inequalities. In geometric probabilities, convex sets play a prominent role, since often the convexity assumptions simplify the situation considerably. Extremal problems for geometric probabilities involving convex bodies ca

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34

Bauer, Christina, and Rolf Schneider. "Extremal problems for geometric probabilities involving convex bodies." Advances in Applied Probability 27, no. 01 (1995): 20–34. http://dx.doi.org/10.1017/s000186780004619x.

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The theory of geometric probabilities is concerned with randomly generated geometric objects. The aim is to compute probabilities of certain geometric events or distributions of random variables defined in a geometric way. Very often the computation even of simple expectations is too difficult, and one has to be satisfied with establishing estimates and, if possible, sharp inequalities. In geometric probabilities, convex sets play a prominent role, since often the convexity assumptions simplify the situation considerably. Extremal problems for geometric probabilities involving convex bodies ca

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35

Boyarchenko, Svetlana. "Irreversible Decisions and Record-Setting News Principles." American Economic Review 94, no. 3 (2004): 557–68. http://dx.doi.org/10.1257/0002828041464452.

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In the now-classical real options theory, the price of an underlying asset is modeled as a geometric Brownian motion, and optimal exercise strategies are described by simple explicit formulas. This paper extends the classical theory to allow any geometric Lévy process to model prices. Such processes may account for fat tails and skewness of probability distributions of commodity prices. The optimal exercise strategies are specified in the paper in terms of statistics of record-setting low or high prices. The formulas derived extend those observed in the Gaussian case, but the form of the resul

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36

Massri, César, Federico Holik, and Ángelo Plastino. "States in generalized probabilistic models: An approach based in algebraic geometry." Mathematica Slovaca 69, no. 1 (2019): 53–70. http://dx.doi.org/10.1515/ms-2017-0202.

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Abstract We present a characterization of states in generalized probabilistic models by appealing to a non-commutative version of geometric probability theory based on algebraic geometry techniques. Our theoretical framework allows for incorporation of invariant states in a natural way.

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37

Angelini, Pierpaolo, and Antonio Maturo. "Geometric interpretation of subjective probability: random numbers and objective conditions of coherence." International Journal of Advanced Statistics and Probability 4, no. 2 (2016): 90. http://dx.doi.org/10.14419/ijasp.v4i2.6319.

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In the domain of the logic of certainty we study the objective notions of the subjective probability with the clear aim of identifying their fundamental characteristics before the assignment, by the individual, of the probabilistic evaluation: probability is an additional and subjective notion that one applies within the range of possibility, thus giving rise to those gradations, more or less probable, that are meaningless in the logic of certainty. When we study the criteria for evaluations under conditions of uncertainty and their corresponding conditions of coherence we show an inevitable d

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38

Harada, Ryo. "Operational and Harmonic-Analytic Aspects of Quasi-Probability Distributions." Open Systems & Information Dynamics 18, no. 03 (2011): 301–20. http://dx.doi.org/10.1142/s1230161211000200.

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Husimi distributions [11] and Wigner distributions [20] are well-known quasi-probability distributions which appear in several contexts. In this paper, we show some remarkable aspects of these distribution functions related to geometric structures of generalized coherent state systems [15,16] and operational quantum physics [7], and a scheme of formulating generalized version of quasi-probability distributions. Our scheme gives concrete formulae of quasi-probability distributions in more direct way from the theory of coherent state systems and clarify their operational meanings, especially of

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39

Mylnikov, A. "Tensor-geometric methods for problems of the circuit theory." Journal of Mathematical Sciences 148, no. 2 (2008): 192–258. http://dx.doi.org/10.1007/s10958-008-0004-5.

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40

Ignatenko, V. F. "Geometric theory of invariants of groups generated by reflections." Journal of Soviet Mathematics 55, no. 6 (1991): 2168–200. http://dx.doi.org/10.1007/bf01095909.

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41

Liu, Bin, Quan Sheng Liu, and Yue Xiu Wu. "On Determination of Geometric Damage Tensor." Advanced Materials Research 97-101 (March 2010): 4293–96. http://dx.doi.org/10.4028/www.scientific.net/amr.97-101.4293.

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It is common to use the damage mechanics to study the deformation of fractured rock mass. The geometric damage tensor is always used to describe the effect of distributed joints on mechanical properties of rock mass. The disadvantages of the traditional determination method about geometric damage is that the natural section of rock mass, mining section and the measurement section are not perpendicular to each other. In this paper, a method combining with the probability theory and field scale line method is adopted to determine the geometric damage tensor. This method is convenient and fast. T

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42

Takenaka, Shigeo. "Integral-geometric construction of self-similar stable processes." Nagoya Mathematical Journal 123 (September 1991): 1–12. http://dx.doi.org/10.1017/s0027763000003627.

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Recently, fractional Brownian motions are widely used to describe complex phenomena in several fields of natural science. In the terminology of probability theory the fractional Brownian motion is a Gaussian process {X(t) : t є R} with stationary increments which has a self-similar property, that is, there exists a constant H (for the Brownian motion H = 1/2, in general 0 < H < 1 for Gaussian processes) called the exponent of self-similarity of the process, such that, for any c > 0, two processes are subject to the same law (see [10]).

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Jughaiman, Abdulaziz. "The Philosophy of Probability Value Behavior: Fractions and Composite Probability Functions in the Continuous Case." European Scientific Journal, ESJ 20, no. 9 (2024): 1. http://dx.doi.org/10.19044/esj.2024.v20n9p1.

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This paper focuses on probability value behavior in the case of continuous sample space by employing fractions intervals and composite functions. The study evaluates value behavior rather than finding values directly, which involves utilization of some concepts from continuity, geometric probability, and measure theory. This paper primarily uses an experiment that contains two major events, head H and tail T, in all their occurrence phases. This spread in infinite and uncountable fractions by a continuous motion within intervals and in the predominant circumstances where events are probabilist

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44

Stojkoski, Viktor, Trifce Sandev, Lasko Basnarkov, Ljupco Kocarev, and Ralf Metzler. "Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing." Entropy 22, no. 12 (2020): 1432. http://dx.doi.org/10.3390/e22121432.

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Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log return

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45

Błaszczyszyn, B., D. Yogeshwaran, and J. E. Yukich. "Limit theory for geometric statistics of point processes having fast decay of correlations." Annals of Probability 47, no. 2 (2019): 835–95. http://dx.doi.org/10.1214/18-aop1273.

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46

Pastorello, Davide. "Geometric Hamiltonian formulation of quantum mechanics in complex projective spaces." International Journal of Geometric Methods in Modern Physics 12, no. 08 (2015): 1560015. http://dx.doi.org/10.1142/s0219887815600154.

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In finite dimension (at least), Quantum Mechanics can be formulated as a proper Hamiltonian theory where a notion of phase space is given by the projective space P(H) constructed on the Hilbert space H of the considered quantum theory. It is well-known P(H) can be equipped with a structure of Kähler manifold, in particular we have a symplectic form and a Poisson structure; Quantum dynamics can be described in terms of a Hamiltonian vector field on P(H). In this paper, exploiting the notion and properties of so-called frame functions, I describe a general prescription for associating quantum ob

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47

Langberg, Naftali A., and David S. Stoffer. "Moving-average models with bivariate exponential and geometric distributions." Journal of Applied Probability 24, no. 1 (1987): 48–61. http://dx.doi.org/10.2307/3214058.

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Two classes of finite and infinite moving-average sequences of bivariate random vectors are considered. The first class has bivariate exponential marginals while the second class has bivariate geometric marginals. The theory of positive dependence is used to show that in various cases the two classes consist of associated random variables. Association is then applied to establish moment inequalities and to obtain approximations to some joint probabilities of the bivariate processes.

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48

Blanchet, J., and P. Glynn. "Uniform renewal theory with applications to expansions of random geometric sums." Advances in Applied Probability 39, no. 4 (2007): 1070–97. http://dx.doi.org/10.1239/aap/1198177240.

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Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results t

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49

Pierpaolo, Angelini. "An Original and Additional Mathematical Model Characterizing a Bayesian Approach to Decision Theory." Journal of Mathematics Research 11, no. 3 (2019): 1. http://dx.doi.org/10.5539/jmr.v11n3p1.

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Abstract:

We propose an original mathematical model according to a Bayesian approach explaining uncertainty from a point of view connected with vector spaces. A parameter space can be represented by means of random quantities by accepting the principles of the theory of concordance into the domain of subjective probability. We observe that metric properties of the notion of $\alpha$-product mathematically fulfill the ones of a coherent prevision of a bivariate random quantity. We introduce fundamental metric expressions connected with transformed random quantities representing changes of origin. We obta

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50

Angelini, Pierpaolo, and Angela De Sanctis. "On a Geometric Extension of the Notion of Exchangeability Referring to Random Events." International Journal of Statistics and Probability 7, no. 2 (2018): 50. http://dx.doi.org/10.5539/ijsp.v7n2p50.

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Abstract:

The notion of exchangeability referring to random events is investigated by using a geometric scheme of representation of possible alternatives. When we distribute among them our sensations of probability, we point out the multilinear essence of exchangeability by means of this scheme. Since we observe a natural one-to-one correspondence between multilinear maps and linear maps, we are able to underline that linearity concept is the most meaningful mathematical concept of probability theory. Exchangeability hypothesis is maintained for mixtures of Bernoulli processes in the same way. We are th

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