Probability Theory Research Articles

Optimizing Betting Strategies with the Kelly Criterion for Economic

This comprehensive study explores the Kelly Criterion, a strategy devised for betting but with significant implications in economic decision-making and wealth management. The criterion, initially developed to maximize long-term growth in independent repeated bets with positive expected net returns, has evolved to encompass a broad range of applications. Through the use of mathematical modeling and probability theory, this research demonstrates how the Kelly Criterion can be effectively applied to enhance wealth accumulation. The societal benefits of this approach are manifold, extending to the reduction of problem gambling rates and promoting responsible gaming. The research also delves into the broader implications of the Kelly Criterion for society, suggesting that its judicious application can significantly shape the future of gambling as a recreational activity. By ensuring informed decision-making, the criterion benefits both individuals and society as a whole, offering a structured approach to managing financial risks and rewards. The expanded scope of this study highlights the versatility of the Kelly Criterion, showing its potential as a pivotal tool in modern economic scenarios, characterized by uncertainty and fluctuating financial markets.

New Combinatorial Identity for the Set of Partitions and Limit Theorems in Finite Free Probability Theory

Abstract We provide a refined combinatorial identity for the set of partitions of $\{1,\dots , n\}$, which plays an important role in investigating several limit theorems related to finite free convolutions. Firstly, we present the finite free analogue of Sakuma and Yoshida’s limit theorem. That is, we provide the limit of $\{D_{1/m}((p_{d}^{\boxtimes _{d}m})^{\boxplus _{d}m})\}_{m\in{\mathbb{N}}}$ as $m\rightarrow \infty $ in two cases: (i) $m/d\rightarrow t$ for some $t>0$ or (ii) $m/d\rightarrow 0$. The second application presents a central limit theorem for finite free multiplicative convolution. We establish a connection between this theorem and the multiplicative free semicircular distributions through combinatorial identities. Our last result gives alternative proofs for Kabluchko’s limit theorems concerning the unitary Hermite and the Laguerre polynomials.

Randomness of Shapes and Statistical Inference on Shapes via the Smooth Euler Characteristic Transform

In this article, we establish the mathematical foundations for modeling the randomness of shapes and conducting statistical inference on shapes using the smooth Euler characteristic transform. Based on these foundations, we propose two chi-squared statistic-based algorithms for testing hypotheses on random shapes. Simulation studies are presented to validate our mathematical derivations and to compare our algorithms with state-of-the-art methods to demonstrate the utility of our proposed framework. As real applications, we analyze a data set of mandibular molars from four genera of primates and show that our algorithms have the power to detect significant shape differences that recapitulate known morphological variation across suborders. Altogether, our discussions bridge the following fields: algebraic and computational topology, probability theory and stochastic processes, Sobolev spaces and functional analysis, analysis of variance for functional data, and geometric morphometrics.

Neutrosophic Topp-Leone Distribution for Interval-Valued Data Analysis

AbstractMany issues in real life are riddled with confusion, vagueness, and ambiguity. The Topp-Leone distribution is a significant one-parameter probability distribution in the context of classical probability theory. There is a gap in the literature when it comes to dealing with circumstances involving interval-valued data with a classical Topp-Leone distribution. So, in this connection, neutrosophic Topp-Leone distribution is presented in this paper as an extension of the traditional Topp-Leone distribution. The new neutrosophic distribution takes into account the indeterminacy and crisp form of interval-valued distributions. The suggested distribution's mathematical features were derived, including moments and related measures, quantile function, survival, hazard, inverted hazard functions, and mills ratio. Maximum likelihood is used to estimate model parameters. Finally, a complex dataset is utilized to show the usefulness of the proposed distributions.

The generalised distribution semantics and projective families of distributions

We generalise the distribution semantics underpinning probabilistic logic programming by distilling its essential concept, the separation of a free random component and a deterministic part. This abstracts the core ideas beyond logic programming as such to encompass frameworks from probabilistic databases, probabilistic finite model theory and discrete lifted Bayesian networks. To demonstrate the usefulness of such a general approach, we completely characterise the projective families of distributions representable in the generalised distribution semantics and we demonstrate both that large classes of interesting projective families cannot be represented in a generalised distribution semantics and that already a very limited fragment of logic programming (acyclic determinate logic programs) in the deterministic part suffices to represent all those projective families that are representable in the generalised distribution semantics at all.

Forecasting technique of transport need for population of cities with high tourist potential during peak seasons

Introduction. One of the key problems in ensuring the quality of recreation for the population is the provision of transport services that meet the transport needs of vacationers. The study of the transport needs of the population of resort cities during the holiday season is caused by the need to assess changing passenger flows as a result of a multiple increase due to vacationers in accordance with their place of attraction and possible adjustments to the routes of regular passenger transportation by road. This determines the relevance of the topic of this article. The purpose of the article is to develop a methodology for determining the needs for transport services for the population and vacationers in cities with high resort potential during the busiest periods, which makes it possible to design transportation processes along regular routes.Materials and methods. General scientific methods of analysis and synthesis, the provisions of probability theory and mathematical statistics, mathematical modeling, as well as the provisions of the technology of transport processes were used as the main research methods. The work carried out the zoning of the city into transport areas with a center of gravity at stopping points of urban passenger transport. Calculations of potential transport needs were carried out on the basis of a survey of vacationers and field observations, which were processed using standard Microsoft Excel and Statistika software.Results. The main result of the work is a methodology for determining the potential transport needs of the population and vacationers in cities with high tourism potential for the development and adjustment of routes for regular passenger transportation by road during the holiday season, which represents the scientific novelty of the research. The technique contains a mathematical model and algorithm.Discussion and conclusion. The use of this methodology will enable the organizer of transportation in resort cities to optimize the organization of regular passenger transportation by road during peak seasons.

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

GeosPIn: A MATLAB library for Geostatistical Petrophysical Inversion

Petrophysical inversion of seismic data is one of the key components of seismic reservoir characterization. The goal of petrophysical inversion is to estimate petrophysical properties of reservoir rocks, such as porosity, volumes of minerals or lithologies, and water and hydrocarbon saturations from seismic data. This process can be performed by combining seismic AVO modeling and rock physics relations with deterministic or probabilistic inverse theory methods, either in a two-step approach based on seismic AVO inversion and rock physics inverse mapping or in a single loop step that includes both. We present a comprehensive MATLAB library named Geostatistical Petrophysical Inversion (GeosPIn). The novelty of the library lies in a single-loop geostatistical inversion method based on the Probability Field Simulation (PFS) and the Ensemble Smoother Multiple Data Assimilation (ES-MDA), and its implementation for applications to 3D datasets. The library also includes a Bayesian petrophysical inversion based on the two-step approach for comparison. We illustrate the main algorithms, explain the structure of the source codes, and demonstrate the library applications through 1D and 3D examples.

Switching particle systems for foraging ants showing phase transitions in path selections

Switching interacting particle systems studied in probability theory are the stochastic processes of hopping particles on a lattice made up of slow and fast particles, where the switching between these types of particles occurs randomly at a given transition rate. This paper explores how such stochastic processes involving multiple particles can model group behaviors of ants. Recent experimental research by the last author’s group has investigated how ants switch between two types of primarily relied cues to select foraging paths based on the current situation. Here, we propose a discrete-time interacting random walk model on a square lattice, incorporating two types of hopping rules. Numerical simulation results demonstrate global changes in selected homing paths, transitioning from trailing paths of the ‘pheromone road’ to nearly optimal paths depending on the switching parameters. By introducing two types of order parameters characterizing the dependence of homing duration distributions on switching parameters, we discuss these global changes as phase transitions in ant path selections. We also study critical phenomena associated with continuous phase transitions.

Estimating the Duration of Construction Works Using Fuzzy Modeling to Assess the Impact of Risk Factors

One of the most pressing issues in the implementation of construction projects is the extension of planned deadlines, significantly impacting project costs. This situation often arises due to inaccurate estimation of construction durations, which rely on normative values without accounting for factors hindering construction progress. Consequently, this article aims to develop an innovative approach for assessing construction durations, considering specific risk factors and their influence on construction activities. Given the difficulty of determining risk factors and their effects during the design phase using classical probability theory, characterized by unknown probability distributions, it is highlighted that this scenario represents planning and implementation under conditions of non-statistical uncertainty. Therefore, the article proposes an approach utilizing elements of fuzzy set theory, particularly fuzzy rules and linguistic variables, to determine delays in individual construction tasks. The proposed approach involves estimating extensions of construction timelines based on a specified probability level of occurrence for risk events and their impact. Additionally, the article provides a theoretical description of the proposed approach and practical calculation examples, demonstrating that the authors’ approach significantly enhances the accuracy of construction timeline forecasts, providing more reliable data for project planning and management.

Encoding and decoding of information in general probabilistic theories

Encoding and decoding of information in general probabilistic theories

Combining Savage and Laplace: a new approach to ambiguity

AbstractThis paper presents a new representation of preference orderings for the study of ambiguity-related decision-making. The central feature is a preference-based decomposition of subjective probabilities that provides information about inherent ambiguity. The probability decomposition is combined with a utility function reflecting the decision-maker’s attitude toward ambiguity. The proposed theory generalizes Savage’s SEU and allows for a straightforward measurement of ambiguity and ambiguity aversion while keeping concepts for measuring risk and risk attitudes unaffected. For the measurement of ambiguity, concepts of probability theory can be used since decision acts can be interpreted as two-dimensional probability distributions. The proposed measure of ambiguity aversion exploits the properties of the utility function in the same way as the Arrow/Pratt measure of risk aversion.

METHODOLOGICAL FEATURES OF TEACHING RANDOM VARIABLES AT THE COMPREHENSIVE SECONDARY SCHOOL

In the evolving landscape of educational methodologies, the integration of complex mathematical concepts such as random variables into secondary school curriculums poses significant challenges and opportunities. As the global educational framework increasingly emphasizes the importance of a robust mathematical foundation, understanding and teaching stochastic elements within the school system has become crucial. This article delves into the nuanced methodological features essential for imparting knowledge of random variables effectively to secondary school students. We explore the historical context of mathematical education reforms in Armenia, highlighting the recent inclusion of stochastic topics in the school programs since 2004. The reformed educational standards introduced in 2020 and implemented in 2023 have further emphasized the need for a structured approach to teaching random variables, alongside other advanced mathematical topics. Through this lens, we examine the preparedness of educators, the adequacy of existing teaching methods, and the potential cognitive hurdles faced by students. Our analysis is rooted in both qualitative assessments and a theoretical framework that aims to bridge the gap between abstract mathematical theories and practical understanding, ensuring that students not only learn but also appreciate the significance of probability theory in everyday contexts .

On the properties of the anisotropic multivariate Hermite-Gauss functions

The Hermite-Gauss basis functions have been extensively employed in classical and quantum optics due to their convenient analytic properties. A class of multivariate Hermite-Gauss functions, the anisotropic Hermite-Gauss functions, arise by endowing the standard univariate Hermite-Gauss functions with a positive definite quadratic form. These multivariate functions admit useful applications in optics, signal analysis and probability theory, however they have received little attention in literature. In this paper, we examine the properties of these functions, with an emphasis on applications in computational optics.

Algebraic groups in non-commutative probability theory revisited

Algebraic groups in non-commutative probability theory revisited

A new approach for the assessment of coastal flooding risk. Application in Rhodes island, Greece

A directional, copula-based, Monte Carlo approach is proposed for the quantitative assessment of a new definition of coastal flooding risk, namely a probability-based index due to extreme combinations of wave conditions and sea levels, which could be a key element for coastal zone planning and management. This approach differs from the standard practice, which focuses on the extreme events and studies either their specific return values or their worst-case scenarios. The adopted approach incorporates mean wave direction as a covariate into the probabilistic representation of sea states, and via the use of a variety of probabilistic and statistical tools, represents a total of five stochastic variables, namely the significant wave height, the peak wave period, the mean wave direction, the storm surge, and tide. Two directional, alternative methods have been developed; a trivariate copula- and a bivariate copula-based method, that seemed to produce quite similar results. Four types of copulas have been examined for their goodness-of-fit, i.e., three Archimedean copulas, namely, the Clayton, the Gumbel, and the Frank copulas, as well as the elliptical Gaussian copula. A comparative analysis of the two methods is performed aiming to point out their pros, as well as their accuracy and complexity issues. The application of the proposed method to the natural beaches of the Rhodes Island in Greece, in combination with a recently developed formula for wave run-up, highlighted the most crucial parameters affecting the coastal flooding risk in the study area.

Probabilistic Analysis of Random Check Intrusion Detection System

The ubiquitous adoption of network-based technologies has left organizations vulnerable to malicious attacks. It has become vital to have effective intrusion detection systems (IDS) that protect the network from attacks. In this paper, we study the intrusion detection problem through the lens of probability theory. We consider a situation where a network receives random malicious signals at discrete time instances, and an IDS attempts to capture these signals via a random check process. We aim to develop a probabilistic framework for intrusion detection under the given scenario. Concretely, we calculate the detection rate of a network attack by an IDS and determine the expected number of detections. We perform extensive theoretical and experimental analyses of the problem. The results presented in this paper would be helpful tools for designing and analyzing intrusion detection systems. We propose a probabilistic framework that could be useful for IDS experts; for a network-based IDS that monitors in real-time, analyzing the entire traffic flow can be computationally expensive. By probabilistically sampling only a fraction of the network traffic, the IDS can still perform its task effectively while reducing the computational cost. However, checking only a fraction of the traffic increases the possibility of missing an attack. This research can help IDS designers achieve appropriate detection rates while maintaining a low false alarm rate. The groundwork laid out in this paper could be used for future research on understanding the probabilities related to intrusion detection.

Sensitivity analysis for matched observational studies with continuous exposures and binary outcomes

Summary Matching is one of the most widely used study designs for adjusting for measured confounders in observational studies. However, unmeasured confounding may exist and cannot be removed by matching. Therefore, a sensitivity analysis is typically needed to assess a causal conclusion’s sensitivity to unmeasured confounding. Sensitivity analysis frameworks for binary exposures have been well-established for various matching designs and are commonly used in various studies. However, unlike the binary exposure case, there still lacks valid and general sensitivity analysis methods for continuous exposures, except in some special cases such as pair matching. To fill this gap in the binary outcome case, we develop a sensitivity analysis framework for general matching designs with continuous exposures and binary outcomes. First, we use probabilistic lattice theory to show our sensitivity analysis approach is finite-population- exact under Fisher’s sharp null. Second, we prove a novel design sensitivity formula as a powerful tool for asymptotically evaluating the performance of our sensitivity analysis approach. Third, to allow effect heterogeneity with binary outcomes, we introduce a framework for conducting asymptotically exact inference and sensitivity analysis on generalized attributable effects with binary outcomes via mixed- integer programming. Fourth, for the continuous outcomes case, we show that conducting an asymptotically exact sensitivity analysis in matched observational studies when both the exposures and outcomes are continuous is generally NP-hard, except in some special cases such as pair matching. As a real data application, we apply our new methods to study the effect of early-life lead exposure on juvenile delinquency. An implementation of the methods in this work is available in the R package doseSens.

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.

Resolvent stochastic processes

A family r λ r_\lambda , λ ∈ C \lambda \in \mathbb {C} , of complex stochastic processes is introduced, which makes it possible to construct a probabilistic representation for the resolvent of the operator − 1 2 d 2 d x 2 -\frac {1}{2}\frac {d^2}{dx^2} . For λ = 0 \lambda =0 , the process r λ r_\lambda coincides with the Brownian local time process.