BESSIS–MOUSSA–VILLANI CONJECTURE AND GENERALIZED GAUSSIAN RANDOM VARIABLES

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

This paper describes a new method for mammographic image noise suppression and enhancement based on the wavelet transform. At each resolution, coefficients associated with noise are modelled by Gaussian random variables; coefficients associated with edges are modelled by generalized Gaussian random variables, and a shrinkage function is assembled based on posterior probabilities. The shrinkage functions at consecutive scales are combined, and then applied to the wavelet coefficients. Given a resolution of analysis, the image denoising process is adaptive (i.e. does not require further parameter adjustments), and the selection of a gain factor provides the desired detail enhancement. The enhancement function was designed to avoid introducing artifacts in the enhancement process, which is essential in mammographic image analysis. Our preliminary results indicate that our method allows us to detect microcalcifications and other suspicious structures in situations where their detection would be difficult otherwise. Compared to other approaches, our method requires less parameter adjustments by the user.

A version of the law of the iterated logarithm is proved for sequences of independent random variables which satisfy the central limit theorem in such a way that the convergence of the appropriate moment-generating functions to that of the standard normal distribution occurs at a particular rate. Kolmogorov's law of the iterated logarithm is a corollary of this theorem which, unlike Kolmogorov's result, does not require boundedness of the random variables. Some iterated logarithm results for weighted averages of independent random variables are shown to follow from the main result. Moreover, some applications to sequences of independent, generalized Gaussian random variables are provided.

Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent or independent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. The approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewed-t distribution and skewed-t Copulas. Thus, our findings are useful for academics, practitioners, and policy makers.

This paper provides a fortran algorithm that can be used to compute the cdf of the product of two normal distribution random variables. We also give references that provide mathematical properties, tables, and applications of this distribution

The literature is replete with acknowledgments that spatial autocorrelation (SA) inflates the variance of a random variable (RV), and that it also may alter other RV distributional properties. In most studies, impacts of SA have been examined only for the three most commonly used distributions: the normal, Poisson (and its negative binomial counterpart), and binomial distributions; much less is known about its effects on two other RVs that are utilized in GIScience research: the beta and the multinomial. The beta distribution—which is considered to be very flexible because it can mimic a uniform, exponential, sinusoidal, and normal RV—can be utilized to analyze the radiance of a remotely sensed image, for example. The multinomial distribution, a generalization of the binomial distribution, has been widely used for land use classification, and to describe land use change. The literature also suggests that RV impacts of negative SA, a neglected topic in spatial analysis, may differ from those of positive SA, at least for some RVs (e.g., the Poisson RV). The purpose of this article is to extend the investigation of effects of SA to beta and multinomial RVs, with both positive SA and negative SA assessed and contrasted with each other, using simulation experiments. The simulation experiments are designed to support this assessment. One of the major discoveries is that impacts of positive SA and negative SA behave similarly when a RV conforms to a normal distribution; however, maximum negative SA is unable to materialize for asymmetric RV, whereas positive SA always converges upon its maximum.

Some strong limit theorems of weighted sums for negatively dependent generalized Gaussian random variables

We establish Berry-Esseen type estimates for random variables with values in a real separable Hilbert space $H$. These estimates are then used to prove the law of the iterated logarithm for sequences of $H$-valued random variables and also to prove a functional form of the law of the iterated logarithm for $H$-valued partial sums as given by Strassen. We also prove a $\\log \\log$ result for $H$-valued symmetric stable random variables.

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

The multilevel Monte Carlo (MLMC) method has been used for a wide variety of stochastic applications. In this paper we consider its use in situations in which input random variables can be replaced by similar approximate random variables which can be computed much more cheaply. A nested MLMC approach is adopted in which a two-level treatment of the approximated random variables is embedded within a standard MLMC application. We analyse the resulting nested MLMC variance in the specific context of an SDE discretisation in which Normal random variables can be replaced by approximately Normal random variables, and provide numerical results to support the analysis.

Target association in sensor data fusion often assumes that both the target tracks and the measurements are described with Gaussian random variables. For some sensor reports, such as passive acoustics, this assumption creates a poor approximation. For the association of measurement to target track, significant errors can occur as a uniform distribution is warped such that the centroid is weighted significantly more than the edges when a single Gaussian is used as the approximation. Using the chi-squared metric to associate a new measurement to an existing target track under this condition can increase the likelihood of association or reduce it significantly. In this paper, a fuzzy-logic approach to data association is enhanced. The original approach showed that it could emulate the chi-squared metric comparing two Gaussian random variables. Here, the approach is enhanced to handle the cases of two uniformly distributed random variables and the case where a uniform measurement is compared to a Gaussian track

<p indent="0mm">The insulation failure of key equipment in power systems caused by gas-solid interface flashover of basin insulators is a key scientific problem that must be solved urgently. The time and space scales of gas electron multiplication and solid-surface charge evolution span multiple orders of magnitude, resulting in an unclear physical mechanism of the gas-solid coupling interface discharge of a basin insulator. A gas-solid coupling surface flashover model was established considering three junction point charge injection, solid surface charge transport, and gas collision ionization. The violent collision ionization in the gas caused by electrons emitted from the solid was taken as the surface flashover criterion. The key parameters characterizing surface charge transport and gas collision ionization were extracted based on gas-solid coupling surface flashover, and a model considering random variables was established to simulate the flashover voltage distribution of epoxy resin. The flashover voltage obeys the Weibull distribution when the charge transport parameters are Gaussian random variables, and the simulation results are consistent with the experimental results. By comparing the simulation and experimental results, it is found that the characteristic flashover voltage and distribution parameter are mainly controlled by the gas and solid media, respectively, and each charge transport random variable has a different effect on the distribution shape parameter. The Weibull distribution of flashover voltage consistent with the experimental results can be obtained by adjusting the standard deviation of charge transport random variables. This research explains the relationship between the Weibull distribution of flashover voltage and the charge transport and gas collision ionization parameters, providing an effective method to simulate the probability distribution characteristics of flashover voltage, which can also be used to evaluate the failure probability of the gas-solid interface under a certain voltage.

In the reliability analysis of safety critical complex engineering structures, a very large number of the system parameters can be considered as random variables. The difficulty in computing the failure probability using the classical first- and second-order reliability methods (FORM and SORM) increases rapidly with the number of variables or ‘dimension’. There are mainly two reasons behind this. The first is the increase in computational time with the increase in the number of random variables. In principle, this problem can be handled with superior computational tools. The second reason, which is perhaps more fundamental, is that there are some conceptual difficulties typically associated with high dimensions. This means that even when one manages to carry out the necessary computations, the application of existing FORM and SORM may still lead to incorrect results in high dimensions. This paper is aimed at addressing this issue. Based on the asymptotic distribution of quadratic form in Gaussian random variables, two formulations for the case when the number of random variables n →∞ is provided. The first is called ‘strict asymptotic formulation’ and the second is called ‘weak asymptotic formulation’. Both approximations result in simple closed-form expressions for the probability of failure of an engineering structure. The proposed asymptotic approximations are compared with existing approximations and Monte Carlo simulations using numerical examples.

ABSTRACTConvolutions of normal random variables often arise in engineering problems, and the probability densities of the sums of these random variables are known in the literature. There are practical situations where specification limits on a process are imposed externally, and the product is typically scrapped if its performance does not fall in the specification range. As such, the actual distribution after inspection is truncated. Despite the practical importance of the role of truncated distributions, there has been little work on the theoretical foundation of convolutions associated with truncated random variables. This is paramount, since convolutions are often used as an important standard in statistical tolerance analysis. In this article, the convolutions of the combinations of truncated normal and truncated skew normal random variables on double and triple truncations are developed. The successful completion of this research task on convolution could help obtain a better understanding of integ...

In 2006, Nadezhkina and Takahashi [N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, {\em SIAM J. Optim.}, 16(4) (2006), 1230-1241.] introduced an iterative algorithm for finding a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality in a real Hilbert space via combining two well-known methods: hybrid and extragradient. In this paper, motivated by Nadezhkina and Takahashi's hybrid-extragradient method we propose and analyze a relaxed hybrid-extragradient method for finding a solution of a constrained convex minimization problem, which is also a common element of the solution set of a variational inclusion and the fixed point set of a strictly pseudocontractive mapping in a real Hilbert space. We obtain a strong convergence theorem for three sequences generated by this algorithm. Based on this result, we also construct an iterative algorithm for finding a solution of the constrained convex minimization problem, which is also a common fixed point of two mappings taken from the more general class of strictly pseudocontractive mappings.