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Improvements in the study of nonparametric maximal exponential models built on Orlicz spaces are proposed. By exploiting the notion of sub-exponential random variable, we give theoretical results which provide a clearer insight into the structure of these models. The explicit constants we obtain when changing the law of Orlicz spaces centered at connected densities allow us to derive uniform bounds with respect to a reference density. [ABSTRACT FROM AUTHOR]

We study a family of representations of the canonical commutation relations (CCR)-algebra, which we refer to as "admissible," with an infinite number of degrees of freedom. We establish a direct correlation between each admissible representation and a corresponding Gaussian stochastic calculus. Moreover, we derive the operators of Malliavin's calculus of variation using an algebraic approach, which differs from the conventional methods. The Fock-vacuum representation leads to a maximal symmetric pair. This duality perspective offers the added advantage of resolving issues related to unbounded operators and dense domains much more easily than with alternative approaches. [ABSTRACT FROM AUTHOR]

The relations between asymptotic stability, the eventual e-property and the e-property of Markov semigroups, acting on measures defined on general (Polish metric) spaces, are studied. While much attention is usually paid to asymptotic stability (for years the e-property has only served as a tool to establish it), it should be noted that the e-property itself is also important as it, e.g., ensures that numerical errors in simulations are negligible. Here, it is shown that any asymptotically stable Markov–Feller semigroup with an invariant measure such that the interior of its support is non-empty satisfies the eventual e-property. Moreover, we prove that any Markov–Feller semigroup, which is strongly stochastically continuous, and which possesses the eventual (Polish metric), also has the e-property. We also present an example highlighting that the assumption of strong stochastic continuity (given in terms of the supremum norm) cannot be relaxed to its weak form, involving pointwise convergence, unless a state space of a process corresponding to a Markov semigroup is a compact metric space. [ABSTRACT FROM AUTHOR]

The search model suggested in this study uses a unit-speed nanosensor to track the fluid's randomly linear flow particle, starting at the real line's origin. The nanosensor oscillates passing through the origin point (towards right and left) in the presence of a sequence of random points, and the particle's position is always changing with a random waiting time that varies with the Gaussian jump length. We may be able to consider the search distance as a function of a discounted effort-reward parameter (factor) because of this uncertainty. We demonstrate analytically how this parameter affects proving the existence of this model and reducing the expected value of the first collision time before the nanosensor returns a particle to the origin. To demonstrate the effectiveness of this model, a numerical example is provided. [ABSTRACT FROM AUTHOR]

We propose a novel personalized concept for the optimal treatment selection for a situation where the response is a multivariate vector that could contain right-censored variables such as survival time. The proposed method can be applied with any number of treatments and outcome variables, under a broad set of models. Following a working semiparametric Single Index Model that relates covariates and responses, we first define a patient-specific composite score, constructed from individual covariates. We then estimate conditional means of each response, given the patient score, correspond to each treatment, using a nonparametric smooth estimator. Next, a rank aggregation technique is applied to estimate an ordering of treatments based on ranked lists of treatment performance measures given by conditional means. We handle the right-censored data by incorporating the inverse probability of censoring weighting to the corresponding estimators. An empirical study illustrates the performance of the proposed method in finite sample problems. To show the applicability of the proposed procedure for real data, we also present a data analysis using HIV clinical trial data, that contained a right-censored survival event as one of the endpoints. [ABSTRACT FROM AUTHOR]

The general framework on the non-local Markovian symmetric forms on weighted lp (p ∈ [ 1 , ∞ ]) spaces constructed by Albeverio et al. (Commn. Math. Phys. 388, 659–706, 2021 Kagawa) by restricting the situation where p = 2, is applied to probability measure spaces describing the space cut-off P(ϕ)2 Euclidean quantum field, the 2-dimensional Euclidean quantum fields with exponential and trigonometric potentials, and the measure associated with the field describing a system of an infinite number of classical particles. For each measure space, the Markov process corresponding to the non-local type stochastic quantization is constructed. [ABSTRACT FROM AUTHOR]

Estimates of best approximations by exponential type analytic functions in Gaussian random variables with respect to the Malliavin derivative in the form of Bernstein–Jackson inequalities with exact constants are established. Formulas for constants are expressed through basic parameters of approximation spaces. The relationship between approximation Gaussian Hilbert spaces and classic Besov spaces are shown. [ABSTRACT FROM AUTHOR]

The article applies mathematical statistics to the dynamic analysis of the demand for talents in large companies. Prediction of the overall demand for talents through the use of deep learning models. We use the theory of diminishing marginal utility to solve the degree of enterprise human resource investment under each index. Based on the analysis of traditional human resource management methods, this paper uses the three-dimensional fuzzy model to construct a new three-dimensional fuzzy model of human resource management model. The research found that the algorithm proposed in this paper can improve the quality of data training and improve the matching of people and posts.

In the paper, we investigate the following fundamental question. For a set K in L 0 (P) , when does there exist an equivalent probability measure Q such that K is uniformly integrable in L 1 (Q) . Specifically, let K be a convex bounded positive set in L 1 (P) . Kardaras (J Funct Anal 266:1913–1927, 2014) asked the following two questions: (1) If the relative L 0 (P) -topology is locally convex on K , does there exist Q ∼ P such that the L 0 (Q) - and L 1 (Q) -topologies agree on K ? (2) If K is closed in the L 0 (P) -topology and there exists Q ∼ P such that the L 0 (Q) - and L 1 (Q) -topologies agree on K , does there exist Q ′ ∼ P such that K is Q ′ -uniformly integrable? In the paper, we show that, no matter K is positive or not, the first question has a negative answer in general and the second one has a positive answer. In addition to answering these questions, we establish probabilistic and topological characterizations of existence of Q ∼ P satisfying these desired properties. We also investigate the peculiar effects of K being positive. [ABSTRACT FROM AUTHOR]

The parametric survival model with Weibull distribution can be used to model a wide range of practical lifetime data. While there have been several studies comparing the fit of various distributions to right-censored and interval censored data, there are no recommendations in the literature on optimal distributions to use for left-censored heavy-tailed data. Parametric Reverse Hazards (PRH) has gained considerable attention from time-to-event data researchers for its excellent properties and appropriateness to analyzing left-censored survival data. To analyze left-censored with heavy-tailed data, we derived the PRH model for a variety of distributions including the Exponential, Log-normal, Inverse Gaussian, Log-logistic, Gompertz–Makeham, Gamma, Generalized Gamma, Inverse Gamma, Generalized Inverse Gamma, Weibull, Inverse Weibull, Generalized Inverse Weibull, Modified Weibull, Flexible Weibull, Power Generalized Weibull, and Marshal–Olkin distributions. Extensive statistical simulations were used to assess the performance of the derived PRH models and compare these to establish a guideline for which distribution/s would "best" fit for left-censored heavy-tailed data. We then applied the best performing model to the South Carolina Enhanced HIV/AIDS Reporting Surveillance System data to explain the effects of different demographic, social, and treatment factors on patients' viral load transition from detectable-to-undetectable levels. [ABSTRACT FROM AUTHOR]

Right-skewed heavy-tailed survival data commonly arise in health-related studies, and the probability distributions proposed to model such survival data are not adequately described in the literature. Although a wide selection of survival distributions is available, applied researchers often face the challenge of choosing a right model for analyzing survival data with these special features in distributional shape. In this article, we evaluated some of the most popular parametric families of distributions as well as nonparametric approaches to model right-censored data using several goodness of fit measures. We carried out extensive simulation studies with different censoring rates to compare the performance of survival models using sixteen different parametric families and the Bernstein polynomials-based nonparametric model to analyze right-skewed heavy-tailed survival data. We also illustrated the use of both parametric and nonparametric approaches to model observed viral load (VL) data from HIV/AIDS patients where we defined the survival outcome as transitioning from an undetectable VL status to a detectable VL status over time. In addition, we used bootstrap samples drawn from the observed VL data and evaluated the estimation properties of all sixteen parametric distributions and the Bernstein polynomial-based method. Finally, results from simulation, observed VL data, and the bootstrap samples were compared using four different divergence-based information criteria. Overall, among the parametric families, the power generalized Weibull model was found to provide adequate fit under several scenarios (censoring rates, choice of information criterion, etc.). The other two families to be competitive with power generalized Weibull were the Burr XII and generalized Gamma distributions. [ABSTRACT FROM AUTHOR]

We give an overview of the concept of random time changes in evolution processes. First of all, we discuss random times in Markov processes. Secondly, we propose to use the concept of random times for dynamical systems. In both cases did appear fractional evolution equations. In the case of Markov processes we arrive to fractional Kolmogorov equations. For dynamical systems it leads to fractional Liouville equations. [ABSTRACT FROM AUTHOR]

In this paper we present some results which generalize the results from Abel and Leviatan (Results Math 75:181–193, 2020), Gavrea and Gavrea (An inequality involving Bernstein polynomials and box-convex functions (submitted)). [ABSTRACT FROM AUTHOR]

We provide a simple proof to a conjecture posed by Raşa (Report of Meeting Conference on Ulam's Type Stability, Rytro, Poland, June 2–6, https://doi.org/10.2478/aupcsm-2014-0011, 2014), and its extension to q-monotone functions obtained by Abel and Leviatan (Results Math 75(4):181–193, 2020). Probabilistic proofs were recently obtained by Komisarski and Rajba (J Math Anal Appl 458:821–830, 2018) and Komisarski and Rajba (Results Math, https://doi.org/10.1007/s00025-021-01417-2, 2021). In this paper, we extend the results from Abel and Leviatan (2020) to the case of box convex functions of order (q, s). [ABSTRACT FROM AUTHOR]

We study the Gaussian noise stability of subsets A of Euclidean space satisfying A = - A . It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. On the other hand, in dimension two and higher, the ball or its complement does not always maximize noise stability among symmetric sets of fixed Gaussian measure. In summary, we provide the first known positive and negative results for the symmetric Gaussian problem. [ABSTRACT FROM AUTHOR]

Restricted isometry property (RIP) provides a near isometric map for sparse signals. RIP of structured random matrices has played a key role for dimensionality reduction and recovery from compressive measurements. In a companion paper, we have developed a unified theory for RIP of group structured measurement operators on generalized sparsity models. The implication of the extended result will be further discussed in this paper in terms of its pros and cons over the conventional theory. We first show that the extended RIP theory enables the optimization of sample complexity over various relaxations of the canonical sparsity model. Meanwhile, the generalized sparsity model is no longer described as a union of subspaces. Thus the sparsity level is not sub-additive. This incurs that RIP of double the sparsity level does not imply RIP on the Minkowski difference of the sparsity model with itself, which is crucial for dimensionality reduction. We show that a group structured measurement operator provides an RIP-like property with additive distortion for non-sub-additive models. This weaker result can be useful for applications like locality-sensitive hashing. Moreover, we also present that the group structured measurements with random sign enables near isometric sketching on any set similar to the Gaussian measurements. Lastly, an extension of theory to infinite dimension is derived and illustrated over selected examples given by Lebesgue measure of support and Sobolev seminorms. [ABSTRACT FROM AUTHOR]

In this paper, we study the diffusive limit of solutions to the generalized Langevin equation (GLE) in a periodic potential. Under the assumption of quasi-Markovianity, we obtain sharp longtime equilibration estimates for the GLE using techniques from the theory of hypocoercivity. We then show asymptotic results for the effective diffusion coefficient in the small correlation time regime, as well as in the overdamped and underdamped limits. Finally, we employ a recently developed numerical method (Roussel and Stoltz in ESAIM Math Model Numer Anal 52(3):1051–1083, 2018) to calculate the effective diffusion coefficient for a wide range of (effective) friction coefficients, confirming our asymptotic results. [ABSTRACT FROM AUTHOR]

Purpose: This study aims to describe the total EEG energy during episodes of intracranial hypertension (IH) and evaluate its potential as a classification feature for IH., New Methods: We computed the sample correlation coefficient between intracranial pressure (ICP) and the total EEG energy. Additionally, a generalized additive model was employed to assess the relationship between arterial blood pressure (ABP), total EEG energy, and the odds of IH., Results: The median sample cross-correlation between total EEG energy and ICP was 0.7, and for cerebral perfusion pressure (CPP) was 0.55. Moreover, the proposed model exhibited an accuracy of 0.70, sensitivity of 0.53, specificity of 0.79, precision of 0.54, F1-score of 0.54, and an AUC of 0.7., Comparison With Existing Methods: The only existing comparable methods, up to our knowledge, use 13 variables as predictor of IH, our model uses only 3, our model, as it is an extension of the generalized model is interpretable and it achieves the same performance., Conclusion: These findings hold promise for the advancement of multimodal monitoring systems in neurocritical care and the development of a non-invasive ICP monitoring tool, particularly in resource-constrained environments., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Authors.)

We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable. [ABSTRACT FROM AUTHOR]

There is now an increasingly large number of proposed concordance measures available to capture, measure and quantify different notions of dependence in stochastic processes. However, evaluation of concordance measures to quantify such types of dependence for different copula models can be challenging. In this work, we propose a class of new methods that involves a highly accurate and computationally efficient procedure to evaluate concordance measures for a given copula, applicable even when sampling from the copula is not easily achieved. In addition, this then allows us to reconstruct maps of concordance measures locally in all regions of the state space for any range of copula parameters. We believe this technique will be a valuable tool for practitioners to understand better the behaviour of copula models and associated concordance measures expressed in terms of these copula models. [ABSTRACT FROM AUTHOR]

In this paper, we derive inequalities for the coefficients in generalized Fourier expansions of (m, n) convex functions in the sense of Popoviciu. Classical Fourier expansions as well as expansions relative to orthogonal polynomials are considered. The results presented here generalize the ones obtained by Niculescu and Rovenţa (Positivity 24(1):129–139, 2020). Some of the results obtained in deriving inequalities for these coefficients can be further used in obtaining Favard-type inequalities similar to the ones given in Wulbert (Math Comput Model 37(12–13):1383–1391, 2003). Favard type inequalities can be used in obtaining probabilistic inequalities which may be further used in fields such as statistical machine learning. [ABSTRACT FROM AUTHOR]

In a number of recent papers, the idea of generalized boundaries has found use in fractal and in multiresolution analysis; many of the papers having a focus on specific examples. Parallel with this new insight, and motivated by quantum probability, there has also been much research which seeks to study fractal and multiresolution structures with the use of certain systems of noncommutative operators; noncommutative harmonic/stochastic analysis. This in turn entails combinatorial, graph operations, and branching laws. The most versatile, of these noncommutative algebras are the Cuntz algebras; denoted O N , N for the number of isometry generators. N is at least 2. Our focus is on the representations of O N. We aim to develop new noncommutative tools, involving both representation theory and stochastic processes. They serve to connect these parallel developments.In outline, boundaries, Poisson, or Martin, are certain measure spaces (often associated to random walk models), designed to encode the asymptotic behavior, e.g., how trajectories diverge when the number of steps goes to infinity. We stress that our present boundaries (commutative or noncommutative) are purely measure-theoretical objects. Although, as we show, in some cases our boundaries may be compared with more familiar topological boundaries. [ABSTRACT FROM AUTHOR]

Given a fixed sigma-finite measure space X , B , ν , we shall study an associated family of positive definite kernels K. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure ν is infinite, but sigma-finite. We introduce such positive definite kernels K · , · with the two variables from the subclass of the sigma-algebra B whose elements are sets with finite ν measure. Our setting and results are motivated by applications. The latter are covered in the second half of the paper. We first make precise the notions of realizations and factorizations for K, and we give necessary and sufficient conditions for K to have realizations and factorizations in L 2 ν . Tools in the proofs rely on probability theory and on spectral theory for unbounded operators in Hilbert space. Applications discussed here include the study of reversible Markov processes, and realizations of Gaussian fields, and their Ito-integrals. [ABSTRACT FROM AUTHOR]

We show that a probability measure on a metric space has full support, if, and only if, the set of all probability measures, that are absolutely continuous with respect to it, is dense in the set of all Borel probability measures. We illustrate the result through a general version of Laplace's method, which in turn leads to general stochastic convergence to global maxima. [ABSTRACT FROM AUTHOR]

A Gaussian copula is widely used to define correlated random variables. To obtain a prescribed Pearson correlation coefficient of ρx between two random variables with given marginal distributions, the correlation coefficient ρz between two standard normal variables in the copula must take a specific value which satisfies an integral equation that links ρx to ρz. In a few cases, this equation has an explicit solution, but in other cases it must be solved numerically. This paper attempts to address this issue. If two continuous random variables are involved, the marginal transformation is approximated by a weighted sum of Hermite polynomials; via Mehler's formula, a polynomial of ρz is derived to approximate the function relationship between ρx and ρz. If a discrete variable is involved, the marginal transformation is decomposed into piecewise continuous ones, and ρx is expressed as a polynomial of ρz by Taylor expansion. For a given ρx, ρz can be efficiently determined by solving a polynomial equation. [ABSTRACT FROM AUTHOR]

We consider reflection-positivity (Osterwalder–Schrader positivity, OS-p.) as it is used in the study of renormalization questions in physics. In concrete cases, this refers to specific Hilbert spaces that arise before and after the reflection. Our focus is a comparative study of the associated spectral theory, now referring to the canonical operators in these two Hilbert spaces. Indeed, the inner product which produces the respective Hilbert spaces of quantum states changes, and comparisons are subtle. We analyze in detail a number of geometric and spectral theoretic properties connected with axiomatic reflection positivity, as well as their probabilistic counterparts; especially the role of the Markov property. This view also suggests two new theorems, which we prove. In rough outline: It is possible to express OS-positivity purely in terms of a triple of projections in a fixed Hilbert space, and a reflection operator. For such three projections, there is a related property, often referred to as the Markov property; and it is well known that the latter implies the former; i.e., when the reflection is given, then the Markov property implies OS-p., but not conversely. In this paper we shall prove two theorems which flesh out a more precise relationship between the two. We show that for every OS positive system E + , θ , the operator E + θ E + has a canonical and universal factorization. Our second focus is a structure theory for all admissible reflections. Our theorems here are motivated by Phillips' theory of dissipative extensions of unbounded operators. The word "Markov" traditionally makes reference to a random walk process where the Markov property in turn refers to past and future: expectation of the future, conditioned by the past. By contrast, our present initial definitions only make reference to three prescribed projection operators, and associated reflections. Initially, there is not even mention of an underlying probability space. This in fact only comes later. [ABSTRACT FROM AUTHOR]

Combining classical arguments for the analysis of the simulated annealing algorithm with the more recent hypocoercive method of distorted entropy, we prove the convergence for large time of the kinetic Langevin annealing with logarithmic cooling schedule. [ABSTRACT FROM AUTHOR]

In this study, we propose a new distribution using the quadratic rank transmutation map named as transmuted two-parameter Lindley distribution (TTLD). This distribution is more flexible than the two-parameter Lindley distribution (TLD). The properties of the TTLD are examined, and estimation methods for the parameters of this distribution are discussed. The usefulness of the TTLD is demonstrated on some real data. [ABSTRACT FROM AUTHOR]

We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form d X t − div γ ( ∇ X t ) d t + β ( X t ) d t ∋ B ( t , X t ) d W t , where γ and β are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on R d and R respectively, and W is a cylindrical Wiener process. Using variational techniques, suitable uniform estimates (both pathwise and in expectation) and some compactness results, well-posedness is proved under the classical Leray–Lions conditions on γ and with no restrictive smoothness or growth assumptions on β . The operator B is assumed to be Hilbert–Schmidt and to satisfy some classical Lipschitz conditions in the second variable. [ABSTRACT FROM AUTHOR]

We characterize when the level sets of a continuous quasi-monotone functional defined on a suitable convex subset of a normed space can be uniquely represented by a family of bounded continuous functionals. Furthermore, we investigate how regularly these functionals depend on the parameterizing level. Finally, we show how this question relates to the recent problem of property elicitation that simultaneously attracted interest in machine learning, statistical evaluation of forecasts, and finance. [ABSTRACT FROM AUTHOR]

For the issue of generating correlated random vector containing discrete variables, one major obstacle is to determine a suitable correlation coefficient ρzin normal space for a specified correlation coefficient ρx. This paper develops a method to solve this problem. First, the double integral evaluated for ρxis transformed into independent standard uniform space, then, a Quasi Monte Carlo method is introduced to calculate the double integral. For a given ρx, an appropriate ρzis determined by a false position method. Compared with existing methodologies, the proposed method is less efficient, but it is relatively easy to implement. [ABSTRACT FROM PUBLISHER]

Rank two parametric perturbations of operators and matrices are studied in various settings. In the finite dimensional case the formula for a characteristic polynomial is derived and the large parameter asymptotics of the spectrum is computed. The large parameter asymptotics of a rank one perturbation of singular values and condition number are discussed as well. In the operator case the formula for a rank two transformation of the spectral measure is derived and it appears to be the t -transformation of a probability measure, studied previously in the free probability context. New transformation of measures is studied and several examples are presented. [ABSTRACT FROM AUTHOR]