Probability Space Research Articles

Path integration method for stochastic responses of differential equations under Lévy white noise.

A path integration (PI) approach that is progressive for studying the stochastic response driven by Lévy white noise is presented. First, a probability mapping is constructed, which decouples the domain of interest for the system state and the probability space derived from the randomness of Lévy white noise within a short time interval. Then, solving the probability mapping yields the short-time response of the system. Finally, the stochastic evolution of the system can be grasped in a stepwise manner based on the fundamental concept of the PI method. The applicability and effectiveness of our approach in addressing the transient and stationary responses under Lévy white noises are verified by Monte Carlo simulation results. Moreover, the advances in utilization of this method are that it eliminates the restriction of the previous PI method on the controlling parameter of Lévy white noises, and it is highly efficient for solving responses of systems under Lévy white noises.

The structure of arbitrary Conze–Lesigne systems

Let Γ \Gamma be a countable abelian group. An (abstract) Γ \Gamma -system X \mathrm {X} - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of Γ \Gamma - is said to be a Conze–Lesigne system if it is equal to its second Host–Kra–Ziegler factor Z 2 ( X ) \mathrm {Z}^2(\mathrm {X}) . The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian Γ \Gamma , namely that they are the inverse limit of translational systems G n / Λ n G_n/\Lambda _n arising from locally compact nilpotent groups G n G_n of nilpotency class 2 2 , quotiented by a lattice Λ n \Lambda _n . Results of this type were previously known when Γ \Gamma was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers U 3 ( G ) U^3(G) norm for arbitrary finite abelian groups G G .

Confidence partitioning sampling filtering

AbstractThe confidence partitioning sampling filter (CPSF) method proposed in this paper is a novel approach for solving the generic nonlinear filtering problem. First, the confidence probability space (CPS) is defined, which restricts the state transition in a bounded and closed state space in the recursive Bayesian filtering. In the posterior CPS, the weighted grid samples, represented the posterior PDF, are obtained by using the partitioning sampling technique (PST). Each weighted grid sample is treated as an impulse function. The approximate expression of the posterior PDF, as key for the PST implementation, is obtained by using the properties of the impulse function in the integral operation. By executing the selection of the CPS and the PST step repeatedly, the CPSF framework is formed to achieve the approximation of the recursive Bayesian filtering. Second, the difficulty of the CPSF framework implementation lies in obtaining the real posterior CPS. Therefore, the space intersection (SI) method is suggested to obtain the approximate posterior CPS. On this basis, the SI_CPSF algorithm, as an executable algorithm, is formed to solve the generic nonlinear filtering problem. Third, the approximate error between the CPSF framework and the recursive Bayesian filter is analyzed theoretically. The consistency of the CPSF framework to the recursive Bayesian filter is proved. Finally, the performances of the SI_CPSF algorithm, including robustness, accuracy and efficiency, are evaluated using four representative simulation experiments. The simulation results showed that SI_CSPF requires far less samples than particle filter (PF) under the same accuracy. Its computation is on average one order of magnitude less than that of the PF. The robustness of the proposed algorithm is also evaluated in the simulations.

Mathematical analysis of a randomized method for fractional Carathéodory type equation with time‐irregular coefficients

AbstractIn this work, we present a novel error analysis for fractional Carathéodory type differential equations with time irregular coefficients. It is based on the filtered probability space, with Banach spaces and then discretized using the randomized numerical method, which is a prototype derived from the Monte Carlo method. We derive error estimates where these estimates also under the low regularity that the function is not assumed to be differentiable. The convergence order is also studied. Finally, we present numerical examples that validate the theoretical conclusions, as well as to highlight the usefulness of our technique and the accuracy of error analysis.

Martingale type, the Gamlen-Gaudet construction and a greedy algorithm

In the present paper we identify those filtered probability spaces (Ω,F,(Fn),P) that determine already the martingale type of a Banach space X. We isolate intrinsic conditions on the filtration (Fn) of purely atomic σ-algebras which determine that the upper ℓp estimates‖f‖Lp(Ω,X)p⩽Cp(‖E(f|F0)‖Lp(Ω,X)p+∑n=1∞‖E(f|Fn)−E(f|Fn−1)‖Lp(Ω,X)p),f∈Lp(Ω,X) imply that the Banach space X is of martingale type p. Our paper complements G. Pisier's investigation [12] and continues the work by S. Geiss and second named author in [3].

Deep learning integrated Bayesian health indicator for cross-machine health monitoring

Constructing a good health indicator (HI) plays a key role in machinery health monitoring. However, due to the training dataset dependency, most conventional deep-learning-based HIs exhibit inconsistent HI scales and limited cross-domain applicability. This study proposes a deep learning integrated Bayesian health indicator (DBHI) to quantify and evaluate system health states in cross-domain applications accurately. The construction of the DBHI comprises two steps: probability space mapping and probabilistic inference. In the first step, the degradation features are extracted from the data and mapped into a probability space by the proposed adversarial autoencoder model. In the second step, the DBHI is calculated by evaluating the relative risk of the current state based on Bayes’ theorem. The effectiveness of the proposed DBHI was validated on two run-to-failure degradation test datasets. In a cross-domain application, the proposed DBHI achieved good performance without retraining compared to other deep learning-based methods that showed invalid results. The proposed DBHI addresses the limitations of conventional deep-learning-based HIs and provides a more reliable and consistent measure of system health, even in a cross-domain application.

Complete convergence for weighted sums of widely negative orthant dependent random variables under the sub-linear expectations

. In this article, we study and establish complete convergence for widely negative orthant dependent random variables under the sub-linear expectations, which extend some complete convergence theorems for widely negative orthant dependent random variables from the classical probability space to sub-linear expectation space.

Criteria for what makes a local optional martingale a true martingale

What makes an optional stochastic exponential a true optional martingale in a probability space where the underlying filtration is not right continuous nor complete. In this paper, we are going to answer this fundamental question. True martingales play an important role in many applications, just consider the large field of the fair pricing of derivative contracts. Therefore, it is important to give a natural, simple and verifiable condition for having a true martingale property in the context of the theory of optional martingales. The way we have done this here is by extending the method of Beneš to local optional martingales, giving a list of conditions for when an optional stochastic exponential of a local optional martingale is a true optional martingale.

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.

Random vector representation of continuous functions and its applications in quantum mechanics

The relation between continuous functions and random vectors is revealed in the paper that the main meaning is described as, for any given continuous function, there must be a sequence of probability spaces and a sequence of random vectors where every random vector is defined on one of these probability spaces, such that the sequence of conditional mathematical expectations formed by the random vectors uniformly converges to the continuous function. This is a random vector representation of continuous functions, which is regarded as a bridge to be set up between real function theory and probability theory. By means of this conclusion, an interesting result about function approximation theory can be obtained. The random vector representation of continuous functions is of important applications in physics. Based on the conclusion, if a large proportion of certainty phenomena can be described by continuous functions and random phenomena can also be described by random variables or vectors, then any certainty phenomenon must be the limit state of a sequence of random phenomena. Then, in the approximation from a sequence of random vectors to a continuous function, the base functions are appropriately selected by us, and an important conclusion for quantum mechanics is deduced: classical mechanics and quantum mechanics are unified. Particularly, an interesting and very important conclusion is introduced as the fact that the mass point motion of a macroscopical object possesses a kind of wave characteristic curve, which is called wave-mass-point duality.

Inverse elastic scattering by random periodic structures

This paper investigates an inverse scattering problem of time-harmonic elastic waves incident on a rigid random periodic structure. A novel and efficient numerical method is proposed to quantify the randomness, i.e., to reconstruct key statistical properties of random structures from boundary measurements of the scattering data. This method consists of two ingredients, the Monte Carlo technique for sampling the probability space and a continuation method with respect to the wavenumber. Unlike existing methods, this approach does not require a priori knowledge of the randomness and can handle a wide range of random structures, including non-Gaussian processes with extremely high levels of complexity. To illustrate the accuracy and effectiveness of the proposed method, numerical examples involving Gaussian and non-Gaussian processes with various covariances are provided.

Invariance of the Mathematical Expectation of a Random Quantity and Its Consequences

Possibility and probability are the two aspects of uncertainty, where uncertainty represents the ignorance of a given individual. The notion of alternative (or event) belongs to the domain of possibility. An event is intrinsically subdivisible and a quadratic metric, whose value is intrinsic or invariant, is used to study it. By subdividing the notion of alternative, a joint (bivariate) distribution of mass appears. The mathematical expectation of X is proved to be invariant using joint distributions of mass. The same is true for X12 and X12…m. This paper describes the notion of α-product, which refers to joint distributions of mass, as a way to connect the concept of probability with multilinear matters that can be treated through statistical inference. This multilinear approach is a meaningful innovation with regard to the current literature. Linear spaces over R with a different dimension can be used as elements of probability spaces. In this study, a more general expression for a measure of variability referred to a single random quantity is obtained. This multilinear measure is obtained using different joint distributions of mass, which are all considered together.

Aggregation of random elements over bounded lattices

Aggregation functions are widely used to fuse information from different sources in a unique value. In many cases, the aggregated information is related to some experimental measure or random sampling of a population. In this direction, it is reasonable to consider aggregation of random elements. In this paper, the concept of aggregation functions of random elements over bounded lattices, which are measurable functions from a probability space to a bounded lattice, is presented. In particular, starting from a partially ordered set, a measurable space is constructed. Random elements are considered to be measurable functions from a probability space to the measurable space. The concept of aggregation of random elements over bounded lattices is defined by generalizing the monotonicity and the boundary conditions in terms of stochastic orders. Several types, such as the induced, random and degenerated aggregations of random elements over bounded lattices are defined and some coherence properties are studied. Particular examples regarding the aggregation of random variables, random graphs and random semi-positive matrices are provided.

Building prescribed quantitative orbit equivalence with the integers

Two groups are orbit equivalent if they both admit an action on a same probability space that share the same orbits. In particular, the Ornstein–Weiss theorem implies that all infinite countable amenable groups are orbit equivalent to the group of integers. To refine this notion between infinite countable amenable groups, Delabie, Koivisto, Le Maître and Tessera introduced a quantitative version of orbit equivalence. They furthermore obtained obstructions to the existence of such equivalence using the isoperimetric profile. In this article, we offer to answer the inverse problem (find a group being orbit equivalent to a prescribed group with prescribed quantification) in the case of the group of integers using the so called Følner tiling shifts introduced by Delabie et al. To do so, we use the diagonal products defined by Brieussel and Zheng giving groups with prescribed isoperimetric profile.

Almost measurable functions on probability spaces

Abstract The notion of (real-valued) almost measurable functions on probability spaces is introduced and some of their properties are considered. It is shown that any almost measurable function may be treated as a quasi-random variable in the sense of [A. Kharazishvili, On some version of random variables, Trans. A. Razmadze Math. Inst. 177 2023, 1, 143–146].

Properties of local orthonormal systems Part I: Unconditionality in Lp$L^p$, 1<p<∞$1<p<\infty$

AbstractAssume that we are given a filtration on a probability space of the form that each is generated by the partition of one atom of into two atoms of having positive measure. Additionally, assume that we are given a finite‐dimensional linear space of ‐measurable, bounded functions on so that on each atom of any ‐algebra , all ‐norms of functions in are comparable independently of or . Denote by the space of functions that are given locally, on atoms of , by functions in and by the orthoprojector (with respect to the inner product in ) onto . Since satisfies the above assumption and is then the conditional expectation with respect to , for such filtrations, martingales are special cases of our setting. We show in this article that certain convergence results that are known for martingales (or rather martingale differences) are also true in the general framework described above. More precisely, we show that the differences form an unconditionally convergent series and are democratic in for . This implies that those differences form a greedy basis in ‐spaces for .

Best Proximity Point Theorems in Non-Archimedean Menger Probabilistic Spaces

In this work, we prove best proximity point theorems for γ-contractions with conditions the weak P-property in non-Archimedean Menger probabilistic metric spaces. We give the notion of γ- proximal contractions of first and second type in non-Archimedean Menger probabilistic metric spaces and also we establish best proximity point theorems for these proximal contractions. Lastly, we complete our study by giving examples that support our results.

Physiological Noise: Definition, Estimation, and Characterization in Complex Biomedical Signals.

Nonlinear physiological systems exhibit complex dynamics driven by intrinsic dynamical noise. In cases where there is no specific knowledge or assumption about system dynamics, such as in physiological systems, it is not possible to formally estimate noise. We introduce a formal method to estimate the power of dynamical noise, referred to as physiological noise, in a closed form, without specific knowledge of the system dynamics. Assuming that noise can be modeled as a sequence of independent, identically distributed (IID) random variables on a probability space, we demonstrate that physiological noise can be estimated through a nonlinear entropy profile. We estimated noise from synthetic maps that included autoregressive, logistic, and Pomeau-Manneville systems under various conditions. Noise estimation is performed on 70 heart rate variability series from healthy and pathological subjects, and 32 electroencephalographic (EEG) healthy series. Our results showed that the proposed model-free method can discern different noise levels without any prior knowledge of the system dynamics. Physiological noise accounts for around 11% of the overall power observed in EEG signals and approximately 32% to 65% of the power related to heartbeat dynamics. Cardiovascular noise increases in pathological conditions compared to healthy dynamics, and cortical brain noise increases during mental arithmetic computations over the prefrontal and occipital regions. Brain noise is differently distributed across cortical regions. Physiological noise is very part neurobiological dynamics and can be measured using the proposed framework in any biomedical series.

A newfangled isolated entropic measure in probability spaces and its applications to queueing theory

<p>It is well established that a diverse range of entropic measures, while remarkably adaptable, must inevitably be complemented by innovative approaches to enhance their effectiveness across various domains. These measures play a crucial role in fields like communication and coding theory, driving researchers to develop numerous new information measures that can be applied in a wide array of disciplines. This paper introduces a pioneering isolated entropic measure and its solicitations to queueing theory the study of dissimilarities of uncertainty. By creating the newly developed discrete entropy, we have articulated an optimization principle where the space capacity is predetermined and solitary evidence accessible is around the mean size. Additionally, we have conveyed the solicitations of "maximum entropy principle" to maximize the entropy probability distributions.</p>

Finding an optimal distribution strategy path in an unpredictable environment

Purpose: This article introduces an innovative method designed to optimize distribution strategies with respect to future uncertainty. It goes beyond the limitations of traditional scenario-based planning that often leads to suboptimal strategies due to the unpredictability of future developments and the challenge of accurately assigning probabilities to these scenarios. Consequently, the method allows selection of the most economically viable future strategy. Methodology: Our methodology diverges from conventional approaches by refraining from making rigid assumptions about the probabilities of future scenarios. Instead, it comprehensively explores the entire allowable probability space to identify an optimal strategy that works well in possible future developments. We employed this method in the case study of a real-world company based in Czechia, where we devised three viable distribution strategies and four model development scenarios. Results: The application of our method demonstrated its effectiveness in selecting the most advantageous strategy, as evidenced by the results of our case study. However, the applicability of the method is contingent upon the accurate definition of potential future scenarios and the evaluation of the performance of different strategies within these scenarios. Conclusion: Our findings suggest that this approach significantly enhances strategic planning under uncertainty. Future research will seek to refine this method further by integrating causal relationships to convey additional information across different model periods, thereby improving the robustness and applicability of the strategy selection process.