Probability Space Research Articles

Uniform syndeticity in multiple recurrence

Abstract The main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers $d,l\geq 1$ and any $\varepsilon> 0$ , we prove the existence of $\delta>0$ and $K\geq 1$ (dependent only on d, l, and $\varepsilon $ ) such that the following holds: Consider a solvable group $\Gamma $ of derived length l, a probability space $(X, \mu )$ , and d pairwise commuting measure-preserving $\Gamma $ -actions $T_1, \ldots , T_d$ on $(X, \mu )$ . Let E be a measurable set in X with $\mu (E) \geq \varepsilon $ . Then, K many (left) translates of $$ \begin{align*} \big\{\gamma\in\Gamma\colon \mu(T_1^{\gamma^{-1}}(E)\cap T_2^{\gamma^{-1}} \circ T^{\gamma^{-1}}_1(E)\cap \cdots \cap T^{\gamma^{-1}}_d\circ T^{\gamma^{-1}}_{d-1}\circ \cdots \circ T^{\gamma^{-1}}_1(E))\geq \delta \big\} \end{align*} $$ cover $\Gamma $ . This result extends and refines uniformity results by Furstenberg and Katznelson. As a combinatorial application, we obtain the following uniformity result. For any integers $d,l\geq 1$ and any $\varepsilon> 0$ , there are $\delta>0$ and $K\geq 1$ (dependent only on d, l, and $\varepsilon $ ) such that for all finite solvable groups G of derived length l and any subset $E\subset G^d$ with $m^{\otimes d}(E)\geq \varepsilon $ (where m is the uniform measure on G), we have that K-many (left) translates of $$ \begin{align*} \{g\in G\colon &m^{\otimes d}(\{(a_1,\ldots,a_n)\in G^d\colon \\ & (a_1,\ldots,a_n),(ga_1,a_2,\ldots,a_n),\ldots,(ga_1,ga_2,\ldots, ga_n)\in E\})\geq \delta \} \end{align*} $$ cover G. The proof of our main result is a consequence of an ultralimit version of Austin’s amenable ergodic Szeméredi theorem.

Kinetic equation for stochastic vector bundles

The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker–Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker–Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention.

Synthesis and Properties of the Ba2PrWO6 Double Perovskite.

We report details on the synthesis and properties of barium praseodymium tungstate, Ba2PrWO6, a double perovskite that has not been synthesized before. Room-temperature (RT) powder X-ray diffraction identified the most probable space group (SG) as monoclinic I2/m, but it was only slightly distorted from the cubic structure. X-ray photoelectron spectroscopy confirmed that the initial (postsynthesis) material contained praseodymium in both 3+ and 4+ charge states. The former (Pr3+) disappeared after exposure to UV light at RT. Photoluminescence studies of Pr3+ revealed that Ba2PrWO6 is an insulator with a band gap exceeding 4.93 eV. Pressure-dependent Raman spectroscopy excluded the possibility of a phase transition up to 20 GPa; however, measurements between 8 and 873 K signified that there might be a change toward the lower symmetry SG below 200 K. Electron paramagnetic resonance spectra revealed the presence of interstitial oxygen which acts as a deep electron trap.

Efficient and Convergent Sequential Pseudo-Likelihood Estimation of Dynamic Discrete Games

Abstract We propose a new sequential Efficient Pseudo-Likelihood (k-EPL) estimator for dynamic discrete choice games of incomplete information. k-EPL considers the joint behaviour of multiple players simultaneously, as opposed to individual responses to other agents’ equilibrium play. This, in addition to reframing the problem from conditional choice probability (CCP) space to value function space, yields a computationally tractable, stable, and efficient estimator. We show that each iteration in the k-EPL sequence is consistent and asymptotically efficient, so the first-order asymptotic properties do not vary across iterations. Furthermore, we show the sequence achieves higher-order equivalence to the finite-sample maximum-likelihood estimator with iteration and that the sequence of estimators converges almost surely to the maximum-likelihood estimator at a nearly superlinear rate when the data are generated by any regular Markov perfect equilibrium, including equilibria that lead to inconsistency of other sequential estimators. When utility is linear in parameters, k-EPL iterations are computationally simple, only requiring that the researcher solve linear systems of equations to generate pseudo-regressors which are used in a static logit/probit regression. Monte Carlo simulations demonstrate the theoretical results and show k-EPL’s good performance in finite samples in both small- and large-scale games, even when the game admits spurious equilibria in addition to one that generated the data. We apply the estimator to analyse competition in the U.S. wholesale club industry.

Ergodic quantum processes on finite von Neumann algebras

Let (M,τ) be a tracial von Neumann algebra with a separable predual and let (Ω,P) be a probability space. A bounded positive random linear operator on L1(M,τ) is a map γ:Ω×L1(M,τ)→L1(M,τ) so that τ(γω(x)a) is measurable for all x∈L1(M,τ) and a∈M, and x↦γω(x) is bounded, positive, and linear almost surely. Given an ergodic T∈Aut(Ω,P), we study quantum processes of the form γTnω∘γTn−1ω∘⋯∘γTmω for m,n∈Z. Using the Hennion metric introduced in [21], we show that under reasonable assumptions such processes collapse to replacement channels exponentially fast almost surely. Of particular interest is the case when γω is the predual of a normal positive linear map on M. As an example application, we study the clustering properties of normal states that are generated by such random linear operators. These results offer an infinite dimensional generalization of the theorems in [21].

A layer assigned probability space partition method for structural small failure probability problem

The Physical Synthesis Method (PSM) stands out as a robust framework for conducting structural reliability analyses due to its clear conceptual foundation. However, this approach often necessitates significant computational resources when addressing scenarios with small failure probabilities. In response to this challenge, this study introduces a layer assigned probability space partition method designed to identify pivotal points based on the ultimate bearing capacity failure criterion of structural components within the PSM framework. Drawing inspiration from Harbitz's β-sphere, this method effectively utilizes the minimum reliability index of components to discern essential representative points within the probability space, thus streamlining computations. The efficacy of this approach is showcased through two case studies: a simply supported beam and a six-story reinforced concrete frame. The outcomes demonstrate that the proposed method, when integrated with PSM, exhibits a substantial enhancement in efficiency compared to the conventional Monte Carlo method. Besides, under equivalent computational resources, it achieves superior computational accuracy compared to the importance sampling method, particularly in scenarios with small failure probabilities. Furthermore, by introducing the notion of a common safe domain, this method addresses challenges in structural reliability analyses involving multiple failure surfaces.

Conditional indicators

. In this paper, we introduce a large class of (so-called) conditional indicators, on a complete probability space with respect to a sub σ-algebra. A conditional indicator is a positive mapping, which is not necessary linear, but may share common features with the conditional expectation, such as the tower property or the projection property. Several characterizations are formulated. Beyond the definitions, we provide some non trivial examples that are used in finance and may inspire new developments in the theory of operators on Riesz spaces.

The information premium on a finite probability space

The information premium on a finite probability space

Improved multi-scale fusion network for solving non-smooth elliptic interface problems with applications

The utilization of deep learning methodologies for addressing partial differential equations (PDEs) has garnered significant attention in recent years. This paper introduces an improved network structure tailored for the discontinuity-capturing, enabling the resolution of interface problem through a unified neural network framework. Employing the probability space filling argument, we show that our model can generate convergent sequences, where the convergence rate depends on the number of sampling points. Several numerical experiments with regular and irregular interfaces are conducted to elucidate the convergence characteristics, thereby validating the theoretical assertions. Furthermore, we apply our approach to effectively solve the size-modified Poisson-Boltzmann test model, utilizing it for predicting electrostatics and the solvation free energies for proteins immersed in ionic solvents, thus showcasing practical applications of our method.

The nonlinear contraction in probabilistic cone b-metric spaces with application to integral equation

The probabilistic cone b-metric space is a novel concept that we describe in this study along with some of its fundamental topological properties and instances. We also established the fixed point theorem for the probabilistic nonlinear Banach contraction mapping on this kind of spaces. Many prior findings in the literature are generalized and unified by our findings. In order to illustrate the basic theorem in ordinary cone b-metric spaces, some related findings are also provided with an application to integral equation.

An ‘elementary’ perspective on reasoning about probability spaces

Abstract This paper is concerned with a two-sorted probabilistic language, denoted by $\textsf{QPL}$, which contains quantifiers over events and over reals, and can be viewed as an elementary language for reasoning about probability spaces. The fragment of $\textsf{QPL}$ containing only quantifiers over reals is a variant of the well-known ‘polynomial’ language from Fagin et al. (1990, Inform. Comput., 87, 78–128). We shall prove that the $\textsf{QPL}$-theory of the Lebesgue measure on $\left [ 0, 1 \right ]$ is decidable, and moreover, all atomless spaces have the same $\textsf{QPL}$-theory. Also, we shall introduce the notion of elementary invariant for $\textsf{QPL}$ and use it to translate the semantics for $\textsf{QPL}$ into the setting of elementary analysis. This will allow us to obtain further decidability results as well as to provide exact complexity upper bounds for a range of interesting undecidable theories.

Deciphering non-Gaussianity of diffusion based on the evolution of diffusivity

Non-Gaussian diffusion of nanoparticles in complex media disrupts Einstein's picture of Brownian motion, and non-Gaussianity is thought to be closely related to diffusing diffusivity generated by spatiotemporal heterogeneities. However, the correlation between non-Gaussianity and the dynamics of heterogeneous environments in anomalous diffusion remains uncertain. Inspired by a recent study by Alexandre [], we demonstrate that non-Gaussianity can be deciphered through the spatiotemporal evolution of heterogeneity-dependent diffusivity distribution. Using diffusion experiments in a linear temperature field and Brownian dynamics simulations, we found that short-time non-Gaussianity can be predicted based on the boundary ratio of the diffusivity distribution; the long-time non-Gaussianity either approaches an asymptotic value of −2 or scales with 1/t, depending on the dominance of particle migration. The temporal variation of non-Gaussianity is determined by an effective Péclet number, which represents a competition between the varying rate of diffusivity and the diffusivity of diffusivity and reveals whether the tail distribution expands or contracts. The tail is more Gaussian than exponential over long times, with exceptions significantly dependent on the diffusivity distribution. Our findings provide a versatile framework for understanding non-Gaussian diffusion in probability space, and shed light on establishing a diffusion spectrum in cells and characterizing nanomedicine transport in biological microenvironment using non-Gaussian statistics. Published by the American Physical Society 2024

Dynamic Programming in Probability Spaces via Optimal Transport

Dynamic Programming in Probability Spaces via Optimal Transport

Unraveling Temporal Dynamics of Multidimensional Statistical Learning in Implicit and Explicit Systems: An X-Way Hypothesis.

Statistical learning enables humans to involuntarily process and utilize different kinds of patterns from the environment. However, the cognitive mechanisms underlying the simultaneous acquisition of multiple regularities from different perceptual modalities remain unclear. A novel multidimensional serial reaction time task was developed to test 40 participants' ability to learn simple first-order and complex second-order relations between uni-modal visual and cross-modal audio-visual stimuli. Using the difference in reaction times between sequenced and random stimuli as the index of domain-general statistical learning, a significant difference and dissociation of learning occurred between the initial and final learning phases. Furthermore, we used a negative and positive occurrence-frequency-and-reaction-time correlation to indicate implicit and explicit learning, respectively, and found that learning simple uni-modal patterns involved an implicit-to-explicit segue, while acquiring complex cross-modal patterns required an explicit-to-implicit segue, resulting in a X-shape crossing of regularity learning. Thus, we propose an X-way hypothesis to elucidate the dynamic interplay between the implicit and explicit systems at two distinct stages when acquiring various regularities in a multidimensional probability space.

Distribution Unified and Probability Space Aligned Teacher-Student Learning for Imbalanced Visual Recognition

Distribution Unified and Probability Space Aligned Teacher-Student Learning for Imbalanced Visual Recognition

Probability density evolution method based stochastic simulation of near-fault pulse-like ground motions

Quantifying the near-fault effect and establishing a reasonable model of near-fault pulse-like ground motions are particularly important for seismic design of structures in near-fault regions. Given the pronounced randomness associated with earthquakes, this study first proposes a novel stochastic model of near-fault pulse-like ground motions by combining the improved finite-fault model (IFFM) and the multivariate copula-based velocity-pulse model (CVPM). Further, a probability density evolution method (PDEM) based stochastic simulation method is developed, by which the model parameters can be determined in a unified probability space so as to ensure the consistency of two independent models. For illustrative purposes, the observed records collected from the 1999 Chi-Chi earthquake are used to generate new stochastic ground motions set. Two ground motions sets based on classical stochastic simulation methods are also presented for comparison. Numerical results show that the proposed method for stochastic simulation of near-fault pulse-like ground motions is reliable; the statistics of peak ground accelerations and spectral characteristics of simulated samples are consistent with station records. Besides, the proposed method accommodates the noteworthy randomness and proportion consistency of components associated with near-fault pulse-like ground motions, making it suitable for the stochastic response and reliability analysis of seismic structures in near-fault regions. This superiority is challenging to classical stochastic simulation methods that lack reasonable consideration of randomness and correlation associated with model parameters.

Hausdorff dimension of certain sets related to random αβ-orbits which are not dense

Let α,β∈[0,1)=R/Z such that at least one of them is irrational. It is known that the random αβ-orbits of the Bernoulli mixture of rotations of α and β by choosing them with equal probability 12 are uniformly distributed modulo 1 with probability one. Chen et al. (2021) [2] showed that the exceptional set in the probability space has full Hausdorff dimension. In this note, we prove that certain sets related to random αβ-orbits which are not dense in R/Z have Hausdorff dimension 1. Our result can be viewed as an improvement of the dimension result obtained by Chen, Wang and Wen.

Group Structure as a Foundation for Entropies.

Entropy can signify different things. For instance, heat transfer in thermodynamics or a measure of information in data analysis. Many entropies have been introduced, and it can be difficult to ascertain their respective importance and merits. Here, we consider entropy in an abstract sense, as a functional on a probability space, and we review how being able to handle the trivial case of non-interacting systems, together with the subtle requirement of extensivity, allows for a systematic classification of the functional form.

Fundamental Theorem of Asset Pricing under fixed and proportional costs in multi-asset setting and finite probability space

The Fundamental Theorem of Asset Pricing is extended to a market model over a finite probability space with many assets that can be exchanged into one another under combined fixed and proportional transaction costs. The absence of arbitrage in this setting is shown to be equivalent to the existence of a family of absolutely continuous single-step probability measures and a multi-dimensional martingale with respect to such a family.

An RBF-LOD Method for Solving Stochastic Diffusion Equations

In this study, we introduce an innovative approach to solving stochastic equations in two and three dimensions, leveraging a time-splitting strategy. Our method combines radial basis function (RBF) spatial discretization with the Crank–Nicolson scheme and the local one-dimensional (LOD) method for temporal approximation. To navigate the probabilistic space inherent in these equations, we employ the Monte Carlo method, providing accurate estimates for expectations and variations. We apply our approach to tackle challenging problems, including two-dimensional convection-diffusion and Burgers’ equations, resulting in reduced computational and memory requirements. Through rigorous testing against diverse problem sets, our methodology demonstrates efficiency and reliability, underscoring its potential as a valuable tool in solving complex multidimensional stochastic equations. We have validated the method’s stability and showcased its convergence as the number of collocation points increases. These findings serve as compelling evidence of the suggested method’s convergence properties.