Probability Measure Research Articles

On Value Distribution of Certain Beurling Zeta-Functions

In this paper, the approximation of analytic functions by shifts ζP(s+iτ) of Beurling zeta-functions ζP(s) of certain systems P of generalized prime numbers is discussed. It is required that the system of generalized integers NP generated by P satisfies ∑m⩽x,m∈N1=ax+O(xδ), a>0, 0⩽δ<1, and the function ζP(s) in some strip lying in σ^<σ<1, σ^>δ, which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces.

Taking Music Seriously: on the Dynamics of 'Mathemusical' Research with a Focus on Hexachordal Theorems

After presenting the general framework of `mathemusical' dynamics, we focus on one music-theoretical problem concerning a special case of homometry theory applied to music composition, namely Milton Babbitt's hexachordal theorem. We briefly discuss some historical aspects of homometric structures and their ramifications in crystallography, spectral analysis and music composition via the construction of rhythmic canons tiling the integer line. We then present the probabilistic generalization of Babbitt's result we recently introduced in a paper entitled ''New hexachordal theorems in metric spaces with probability measure'' and illustrate the new approach with original constructions and examples.

Approximation of splines in Wasserstein spaces

This paper investigates a time discrete variational model for splines in Wasserstein spaces to interpolate probability measures. Cubic splines in Euclidean space are known to minimize the integrated squared acceleration subject to a set of interpolation constraints. As generalization on the space of probability measures the integral of the squared acceleration is considered as a spline energy and regularized by addition of the usual action functional. Both energies are then discretized in time using local Wasserstein-2 distances and the generalized Wasserstein barycenter. The existence of time discrete regularized splines for given interpolation conditions is established. On the subspace of Gaussian distributions, the spline interpolation problem is solved explicitly and consistency in the discrete to continuous limit is shown. The computation of time discrete splines is implemented numerically, based on entropy regularization and the Sinkhorn algorithm. A variant of Nesterov’s accelerated gradient descent algorithm is applied for the minimization of the fully discrete functional. A variety of numerical examples demonstrate the robustness of the approach and show striking characteristics of the method. As a particular application the spline interpolation for synthesized textures is presented.

An effective theoretical and experimental analysis method for the improved slime mould algorithm

Metaheuristic intelligent optimization algorithms are effective methods for solving high-dimensional nonlinear complex optimization problems. The slime mould algorithm is a novel intelligent optimization algorithm proposed in 2020. However, the basic slime mould algorithm still has some shortcomings, such as slow convergence rate, easy to fall into local extremum, and unbalanced exploration and development capability. To further improve and expand the optimization ability and application scope of the slime mould algorithm, and enhance its performance in solving large-scale complex optimization problems, this paper proposes a slime mould algorithm (PPMSMA) based on Gaussian perturbation and phased position update, positive variation, and multi-strategy greedy selection. Firstly, Gaussian perturbation and phased position update mechanism are introduced to avoid the difficulty of the algorithm to jump out of the local extrema and also to speed up the convergence of the algorithm. Then, a positive variation strategy based on the sine cosine mechanism is introduced to move the variation of the population towards a better direction. Finally, a multi-strategy greedy selection mechanism is introduced, which effectively improves the search ability of the algorithm. The analysis and research on the optimization ability and performance of metaheuristic algorithms mainly include two aspects: theoretical analysis and experimental testing. Theoretical analysis has always been a relatively weak link in the research of metaheuristic algorithms, and there is currently no clear and effective method formed. For experimental testing, although there are more methods, they often lack systematization and adequacy. In this paper, a more complete, fine-grained and systematic approach to theoretical and experimental analysis is proposed. In the theoretical analysis part, the time complexity and spatial complexity of the PPMSMA algorithm are analytically proved to be the same as the basic slime mould algorithm, and the probability measure method is used to prove that PPMSMA algorithm can converge to the global optimal solution. In the simulation experiment section, the PPMSMA algorithm is compared with multiple sets of 10 representative comparison algorithms on the CEC2017 complex test function set suite for optimization accuracy analysis, Friedman comprehensive ranking analysis, average optimization rate analysis of PPMSMA relative to other algorithms, convergence curve analysis, and Wilcoxon rank-sum test analysis. To further examine the scalability of the improved algorithm in solving large-scale optimization problems, PPMSMA is compared with the above 10 comparative algorithms under 1000 dimensional conditions in the large-scale global optimization test set CEC2010, and the solution stability of each algorithm is analyzed through violin plots. The results show that the PPMSMA algorithm has significantly improved convergence performance, optimization accuracy, and solution stability in both high-dimensional and large-scale complex problems, and has significant advantages compared to multiple sets of 10 representative comparative algorithms. Finally, PPMSMA and 10 other comparative algorithms are used to solve engineering design optimization problems with different complexities. The experimental results validate the universality, reliability, and superiority of PPMSMA in handling engineering design constraint optimization problems.

Stochastic modelling of in-structure concrete strength based on stochastic damage model and Burr distribution

The probability measure of low-quality concrete is essential for the reliability analysis of concrete structures. However, this problem is usually neglected, and the normal distribution or lognormal distribution is often selected as the probability distribution of concrete strength. In this study, a better solution for this problem is given by theoretically deriving of the Burr distribution based on the stochastic damage model. A large amount of in-situ test data in engineering structures is applied to perform a K-S test of different distribution types and to fit the distribution parameters. As a result, the advantage of Burr distribution in representing the tail probability is explained by both theoretical derivation and fitting results. And the Burr distribution is accepted by the K-S test in every strength grade while the other distribution types are all partly rejected.

Probabilistic deconstruction of a theory of gravity, part II: Curved space

We propose that the underlying context of holographic duality and the Ryu-Takayanagi formula is that the volume measure of spacetime is a probability measure constrained by quantum dynamics. We define quantum stochastic processes using joint quantum distributions which are realized in a quantum system as expectation values of products of projectors. In anti-de Sitter JT gravity, we show that Einstein’s equations arise from the evolution of probability under the quantum stochastic process induced by the boundary, with the area of compactified space in the gravitational theory identified as a probability density evolving under the quantum process. Extrapolating these and related results in flat JT gravity found in [SciPost Phys. 15, 174 (2023)], we conjecture that general relativity arises in the semi-classical limit of the evolution of probability with respect to quantum stochastic processes.

The variational principle for probabilistic measure and Hashin–Shtrikman bounds

The paper is a review of Hashin–Shtrikman type bounds for effective moduli of conductivity and elasticity of polycrystals and composites written from the perspective of the variational principle for probabilistic measure. The results for such bounds are rederived in probabilistic terms. Remarkably, in probabilistic terms the Hashin–Shtrikman approach gets especially simple form. Besides, a clear distinction arises between the basic assumption, the choice of the trial field, and the simplifying assumptions, like geometrical isotropy, physical isotropy, texture isotropy, etc. We filled out several gaps. First, we derive an integral equation to be solved to get the bounds when the simplifying assumptions do not hold. Second, we extend the bounds for polycrystals with the cubic symmetry of crystallites to all thermodynamically possible crystallites; previously such bounds were found for crystallites with special elastic properties. One practical outcome considered is the derivation of approximate formulae for the temperature dependence of effective elastic moduli. Third, for crystallites with non-cubic symmetries, we formulated algebraic variational problems to be solved numerically to obtain the bounds, and solved these problems for several materials.

Probability measures on path space for rectilinear damped pressureless Euler-Poisson equations

We show global existence of a measure-valued solution to the Cauchy problem for the rectilinear Euler-Poisson equations, which govern flow of pressureless dust experiencing linear damping in the gravitational field generated by its own mass, and a given background charge field modeling exterior gravitational force. The velocity of the dust is a Borel measurable function, and the distribution of mass is given by time-dependent probability measures on the real line. Upon adapting the method established in Hynd (2020) [25], we obtain the solution by construction of a probability measure on the path space, through approximation of finite sticky particles.

New fractal dimensions of measures and decompositions of singularly continuous measures

By introducing a novel expression for the Hausdorff and packing measures with respect to gauge functions, we establish a connection between different approaches to defining the exact Hausdorff and packing dimensions of measures. Furthermore, we define the upper and lower Hausdorff and packing dimensions of a Borel probability measure and demonstrate that these dimensions can be represented using an entropy formula. This paper also addresses an intriguing question raised multiple times regarding the structural representation of singular probability measures. We provide an affirmative response to this question by utilizing characteristic measures.

Efficient optimization in stochastic production planning problems with product substitution

We consider the stochastic production planning problem with product substitution, which can be decomposed into several optimization subproblems with sequential decisions. The decision variables in each time period include (1) the product substitution decision and (2) the recipe input quantity decision. The goal is to minimize the total of production cost, holding cost, and shortage cost, while achieving a service level for demand satisfaction.Since this optimization model involves analytically intractable probabilistic formulation, traditional mathematical programming techniques cannot be readily applied. We develop the deterministic SPLINE and the stochastic R-SPLINE algorithms for different scenarios. The probability generating function is embedded into the deterministic algorithm to exactly calculate the desired performance measures, which is reasonable when dealing with independent data (with a small number of product classes as well). The stochastic R-SPLINE algorithm uses simulation to estimate the desired probabilistic measures, allowing correlations between different production recipes as well as between different demand classes. We also present a convergence analysis for the stochastic R-SPLINE algorithm. Experimental results demonstrate the efficiency of the developed algorithms compared to other existing approaches.

Topological amenability of semihypergroups

Abstract In this article, we introduce and explore the notion of topological amenability in the broad setting of (locally compact) semihypergroups. We acquire several stationary, ergodic and Banach algebraic characterizations of the same in terms of convergence of certain probability measures, total variation of convolution with probability measures and translation of certain functionals, as well as the F-algebraic properties of the associated measure algebra. We further investigate the interplay between restriction of convolution product and convolution of restrictions of measures on a sub-semihypergroup. Finally, we discuss and characterize topological amenability of sub-semihypergroups in terms of certain invariance properties attained on the corresponding measure algebra of the parent semihypergroup. This in turn provides us with an affirmative answer to an open question posed by J. Wong in 1980.

Analysis of the economic effects on biopsychosocial homeostasis from a three-dimensional probability measurement system

Analysis of the economic effects on biopsychosocial homeostasis from a three-dimensional probability measurement system

Capacities and Choquet averages of ultrafilters

We show that a normalized capacity ν : P ( N ) → R \nu : \mathcal {P}(\mathbf {N})\to \mathbf {R} is invariant with respect to an ideal I \mathcal {I} on N \mathbf {N} if and only if it can be represented as a Choquet average of { 0 , 1 } \{0,1\} -valued finitely additive probability measures corresponding to the ultrafilters containing the dual filter of I \mathcal {I} . This is obtained as a consequence of an abstract analogue in the context of Archimedean Riesz spaces.

Conditional McKean-Vlasov SDEs with jumps and Markovian regime-switching: Wellposedness, propagation of chaos, averaging principle

The conditional McKean-Vlasov SDEs with jumps and Markovian regime-switching are investigated in this work. We use the L2-Wasserstein distance to measure the regularity of the coefficients in the probability measure argument. Also, we establish the propagation of chaos for the associated mean-field interaction particle system with common noise and provide an explicit bound on the convergence rate. Furthermore, an averaging principle is established for two time-scale conditional McKean-Vlasov equations, where much attention is paid to the convergence of the conditional distribution term.

Informational Updates and the Derivative Pricing Kernel

It is common in financial markets for market makers to offer prices on derivative instruments even though they are uncertain about the underlying asset’s value. This paper studies the mathematical problem that arises as a result. Derivatives are priced in the risk-neutral framework, so as the market maker acquires more information about the underlying asset, the change of measure for transition to the risk-neutral framework (the pricing kernel) evolves. This evolution takes a precise form when the market maker is Bayesian. It is shown that Bayesian updates can be characterized as additional informational drift in the underlying asset’s stochastic process. With Bayesian updates, the change of measure needed for pricing derivatives is two-fold: the first change is from the prior probability measure to the posterior probability measure, and the second change is from the posterior probability measure to the risk-neutral measure. The relation between the regular pricing kernel and the pricing kernel under this two-fold change of measure is characterized.

Optimal investment–consumption–insurance strategy with inflation risk and stochastic income in an Itô–Lévy setting

This paper’s focus is on finding the optimal strategies for a trader who invests in stock, a money market account and an inflation-linked index bond. The stock follows a jump diffusion process and the bond is linked to inflation making the two risky. The optimal strategies are determined on two generations of the life of an investor, that is before the investor dies and after the investor dies. We applied the concept of change of probability measures considering Girsanov’s and the Radon–Nikodym theorems. We found the generator of the Backward Stochastic differential equations defined and employed the Hamilton–Jacobi–Bellman (HJB) dynamic programming in finding the stochastic optimal controls of interest.

Communications to the Pseudo-Additive Probability Measure and the Induced Probability Measure Realized by <img src=image/13492447_01.gif>

Communications to the Pseudo-Additive Probability Measure and the Induced Probability Measure Realized by <img src=image/13492447_01.gif>

Statistical inference with regularized optimal transport

Abstract Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning and applied mathematics. However, OT distances suffer from computational and statistical scalability issues to high dimensions, which motivated the study of regularized OT methods like slicing, smoothing and entropic penalty. This work establishes a unified framework for deriving limit distributions of empirical regularized OT distances, semiparametric efficiency of the plug-in empirical estimator and bootstrap consistency. We apply the unified framework to provide a comprehensive statistical treatment of (i) average- and max-sliced $p$-Wasserstein distances, for which several gaps in existing literature are closed; (ii) smooth distances with compactly supported kernels, the analysis of which is motivated by computational considerations; and (iii) entropic OT, for which our method generalizes existing limit distribution results and establishes, for the first time, efficiency and bootstrap consistency. While our focus is on these three regularized OT distances as applications, the flexibility of the proposed framework renders it applicable to broad classes of functionals beyond these examples.

Functor of Idempotent Probability Measures with Compact Support and Open Mappings

Functor of Idempotent Probability Measures with Compact Support and Open Mappings

SUBSETS OF POSITIVE AND FINITE MULTIFRACTAL MEASURES

Sets of infinite multifractal measures are awkward to work with, and reducing them to sets of positive finite multifractal measures is a very useful simplification. The aim of this paper is to show that the multifractal Hausdorff measures satisfy the “subset of positive and finite measure” property. We apply our main result to prove that the multifractal function dimension is defined as the supremum over the multifractal dimension of all Borel probability measures.