Gaussian Random Variables Research Articles

Two problems on random analytic functions in Fock spaces

AbstractLet $f(z)=\sum _{n=0}^\infty a_n z^n$ be an entire function on the complex plane, and let ${\mathcal R} f(z) = \sum _{n=0}^\infty a_n X_n z^n$ be its randomization induced by a standard sequence $(X_n)_n$ of independent Bernoulli, Steinhaus, or Gaussian random variables. In this paper, we characterize those functions $f(z)$ such that ${\mathcal R} f(z)$ is almost surely in the Fock space ${\mathcal F}_{\alpha }^p$ for any $p, \alpha \in (0,\infty )$ . Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, also known as regularity improvement under randomization within the scale of Fock spaces. Other results obtained in this paper include: (a) a characterization of random analytic functions in the mixed-norm space ${\mathcal F}(\infty , q, \alpha )$ , an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; and (c) a complete description of random multipliers between different Fock spaces.

High-dimensional asymptotic expansion of the null distribution for Schott’s test statistic for complete independence of normal random variables

This article is concerned with the testing complete independence for the elements of observed vector. Schott proposed the testing statistic T and gave limiting null distribution under the high-dimensional asymptotic framework that the sample size n and the dimensionality p go to infinity together while p/n converges to a positive constant. In this article we give a one-term asymptotic expansion of the null distribution for T as min{n,p} tends toward infinity. We derive a correction of the critical point for Schott’s test based on this expansion. The finite sample size and dimensionality performance for attained significance level is evaluated in a simulation study and the results are compared to those of Schott’s test.

An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

On Minimax Detection of Gaussian Stochastic Sequences with Imprecisely Known Means and Covariance Matrices

We consider the problem of detecting (testing) Gaussian stochastic sequences (signals) with imprecisely known means and covariance matrices. An alternative is independent identically distributed zero-mean Gaussian random variables with unit variances. For a given false alarm (1st-kind error) probability, the quality of minimax detection is given by the best miss probability (2nd-kind error probability) exponent over a growing observation horizon. We study the maximal set of means and covariance matrices (composite hypothesis) such that its minimax testing can be replaced with testing a single particular pair consisting of a mean and a covariance matrix (simple hypothesis) without degrading the detection exponent. We completely describe this maximal set.

An adaptive PC-Kriging method for time-variant structural reliability analysis

The practical application of time-variant reliability analysis is limited by its computationally expensive models which describe the structural system behavior. This paper presents a new adaptive PC-Kriging (APCK) approach to accurately and efficiently assess the time-variant reliabilities. Time interval is firstly discretized with a series of time instants and then the stochastic process is reconstructed by standard normal random variables and deterministic function of time. PC-Kriging (PCK) models are built at each time instant to predict the instantaneous responses of performance function. To improve the accuracy and efficiency, a new update strategy based on the integration of U- and H- learning functions is developed to refine the PCK models of instantaneous responses. One or two best samples are identified by the proposed learning criterion for updating the PCK models. Finally, Monte Carlo simulation (MCS) is used to estimate the time-variant reliability based on the updated PCK models. Four examples are used to validate the accuracy and efficiency of the proposed method.

Technical Note—Masking Anstreicher’s linx Bound for Improved Entropy Bounds

A fundamental NP-hard combinatorial-optimization in the area of statistical designs is the maximum-entropy sampling problem (MESP), which seeks to maximize Shannon's “differential entropy” over all subsets of a prespecified cardinality from a set of n Gaussian random variables. This problem has applications in many areas, such as the redesign of environmental-monitoring networks. Most algorithms for exact solution of MESP are branch-and-bound based, and one of the best upper bounds is based on Anstrecher's recent concave “linx relaxation” of differential entropy. A key paradigm for improving bounds is by “masking” the covariance of the random variables with a correlation matrix. The main result establishes that in the best case, the linx bound can be improved by an amount that is at least linear in n by masking. These and other recent results on the hot topic of MESP are leading to practical algorithms for exact solution of meaningful design problems in applied areas such as environmental statistics.

Enhanced memory-type charts for monitoring the ratio of two normal random variables

It is common in the SPC literature to enhance the sensitivity of a control chart using an auxiliary information based estimator of the underlying process parameter(s) that requires information on the study variable, as well as supplementary information on one or more auxiliary variables. In this study, we develop a novel approach to monitor the ratio of two normal means when the underlying process follows a multivariate normal distribution. Under the proposed setup, to monitor the process ratio, the supplementary information can be utilized on one or more auxiliary variables. In addition, we propose the EWMA and CUSUM charts for monitoring the process ratio of the means. Using Monte Carlo simulations, the run length profiles of these control charts are computed, and then evaluated in terms of the run length properties. It is found that the proposed charts may uniformly and substantially outperform the existing charts when detecting upward and downward shifts simultaneously in the process ratio of the means. A real dataset is also considered to demonstrate the implementation of existing and proposed charts.

Some new Gaussian product inequalities

The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality E[X12X22m2X32m3]≥E[X12]E[X22m2]E[X32m3] for any centered Gaussian random vector (X1,X2,X3) and m2,m3∈N. First, we show that this inequality is implied by a combinatorial inequality. The combinatorial inequality can be verified directly for small values of m2 and arbitrary m3. Hence the corresponding cases of the three-dimensional inequality are proved. Second, we show that the three-dimensional inequality is equivalent to an improved Cauchy-Schwarz inequality. This observation leads us to derive some novel moment inequalities for bivariate Gaussian random variables.

Modeling Gaussian and Gamma random fields for layered material properties with transitional zones

Layered geomaterials (e.g., soil and rock) are common in nature due to the long-term geological sedimentation. For such materials, both the inter- and intra-layer spatial variability of properties need to be considered in geotechnical engineering. In this study, efficient and simple methods are proposed to generate multi-layer random fields based on the additive property of Gaussian and Gamma random variables. The multi-layer random field is capable of simultaneously characterizing the inter- and intra-layer spatial variability of layered material properties. Moreover, smooth transitional zones between different layers are automatically formed, which can guarantee the continuity of layered material properties. The auto-correlation function of the multi-layer random field decreases monotonically with distance due to the nature of the proposed methods, which is suitable to simulate material properties. Furthermore, a seepage evaluation of three-dimensional concrete dam foundation with layered soil is performed to verify the feasibility of the proposed methods in practical applications. The application example highlights the superiorities of the proposed methods in simulating layered material properties.

Monitoring the Ratio of Two Normal Variables Based on Triple Exponentially Weighted Moving Average Control Charts with Fixed and Variable Sampling Intervals

In statistical process control (SPC), the ratio of two normal random variables (RZ) is a valuable statistical indicator to be taken as the charting statistic. In this work, we propose a triple exponentially weighted moving average (TEWMA) chart for monitoring the RZ. Additionally, the variable sampling interval (VSI) strategy has been adopted to different control charts by researchers. With the application of this strategy, the VSI-TEWMA-RZ chart is then developed to further improve the performance of the proposed TEWMA-RZ chart. The run length (RL) properties of the proposed TEWMA-RZ and VSI-TEWMA-RZ charts are obtained by the widely used Monte-Carlo (MC) simulations. Through the comparisons with the VSI-EWMA-RZ and the VSI-DEWMA-RZ charts, the VSI-TEWMA-RZ chart is statistically more sensitive than the VSI-EWMA-RZ and the VSI-DEWMA-RZ charts in detecting small and moderate shifts. Moreover, it turned out that the VSI-TEWMA-RZ chart has better performance than the TEWMA-RZ chart on the whole. Furthermore, this paper illustrates the implementation of the proposed charts with an example from the food industry.

Non-Gaussianities in the statistical distribution of heavy OPE coefficients and wormholes

The Eigenstate Thermalization Hypothesis makes a prediction for the statistical distribution of matrix elements of simple operators in energy eigenstates of chaotic quantum systems. As a leading approximation, off-diagonal matrix elements are described by Gaussian random variables but higher-point correlation functions enforce non-Gaussian corrections which are further exponentially suppressed in the entropy. In this paper, we investigate non- Gaussian corrections to the statistical distribution of heavy-heavy-heavy OPE coefficients in chaotic two-dimensional conformal field theories. Using the Virasoro crossing kernels, we provide asymptotic formulas involving arbitrary numbers of OPE coefficients from modular invariance on genus-g surfaces. We find that the non-Gaussianities are further exponentially suppressed in the entropy, much like the ETH. We discuss the implication of these results for products of CFT partition functions in gravity and Euclidean wormholes. Our results suggest that there are new connected wormhole geometries that dominate over the genus-two wormhole.

Distributions of Polynomials in Gaussian Random Variables under Constraints on the Powers of Variables

Distributions of Polynomials in Gaussian Random Variables under Constraints on the Powers of Variables

On the numerical computation of Fisher–Rao based distances and KL-divergences between Gaussian random fields

Random fields are among the mathematical models used in the study of non-deterministic and non-linear complex systems. A fundamental aspect in the characterization of such models is to define intrinsic properties and to understand how these measures change over time. In this paper, we propose an information-geometric framework to analyze Gaussian–Markov random fields (GMRF’s) by the definition of a Fisher information based space. With this approach, it is possible to measure the variations in each component of the metric tensor that equips the underlying parametric space, when visiting different entropic states. Using Markov Chain Monte Carlo simulations, we propose a method based on infinitesimal displacements to compute distances between two systems operating in different regimes using the Fisher metric. Moreover, we derive an expression for the KL-divergence (relative entropy) between two GMRF models, showing that, it symmetrized version works reasonably well as a replacement for the Fisher information based distance. Finally, information cycles that relate components of the metric tensor and the system’s entropy reveal an asymmetric pattern of evolution when the system moves towards different entropic states, indicating that the simple interaction between several Gaussian random variables can lead to the emergence of an intrinsic notion of time in the evolution of a random field, based in the geometric properties of its parametric space.

Optimization landscape in the simplest constrained random least-square problem

We analyze statistical features of the ‘optimization landscape’ in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of a system of M linear equations in N unknowns: ( a k , x ) = b k , k = 1, …, M on the N-sphere x 2 = N. We treat both the N-component vectors a k and parameters b k as independent mean zero real Gaussian random variables. First, we derive the exact expressions for the mean number of stationary points of the least-square loss function in the overcomplete case M > N in the framework of the Kac-Rice approach combined with the random matrix theory for Wishart ensemble. Then we perform its asymptotic analysis as N → ∞ at a fixed α = M/N > 1 in various regimes. In particular, this analysis allows to extract the large deviation function for the density of the smallest Lagrange multiplier λ min associated with the problem, and in this way to find its most probable value. This can be further used to predict the asymptotic mean minimal value of the loss function as N → ∞. Finally, we develop an alternative approach based on the replica trick to conjecture the form of the large deviation function for the density of at N ≫ 1 and any fixed ratio α = M/N > 0. As a by-product, we find the compatibility threshold α c < 1 which is the value of α beyond which a large random linear system on the N-sphere becomes typically incompatible.

The basic distributional theory for the product of zero mean correlated normal random variables

AbstractThe product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener‐Itô integrals.

On the Maximum Domain of Attraction for Transformations of a Normal Random Variable

Limit distributions of the maximum of independent copies of a transformation of a Gaussian random variable are studied. Sufficient and necessary conditions are found for the transformations belonging to Fréchet and Weibull maximum domains of attraction. Simple sufficient conditions are also given.

Directed Gaussian graphical models with toric vanishing ideals

Directed Gaussian graphical models are statistical models that use a directed acyclic graph (DAG) to represent the conditional independence structures between a set of jointly normal random variables. The DAG specifies the model through recursive factorization of the parametrization, via restricted conditional distributions. In this paper, we make an attempt to characterize the DAGs whose vanishing ideals are toric ideals. In particular, we give some combinatorial criteria to construct such DAGs from smaller DAGs which have toric vanishing ideals. An associated monomial map called the shortest trek map plays an important role in our description of toric Gaussian DAG models. For DAGs whose vanishing ideal is toric, we prove results about the generating sets of those toric ideals.

Cauchy, normal and correlations versus heavy tails

A surprising result of Pillai and Meng (2016) showed that a transformation ∑j=1nwjXj/Yj of two iid centered normal random vectors, (X1,…,Xn) and (Y1,…,Yn), n>1, for any weights 0≤wj≤1, j=1,…,n, ∑j=1nwj=1, has a Cauchy distribution regardless of any correlations within the normal vectors. The correlations appear to lose out in the competition with the heavy tails. To clarify how extensive this phenomenon is, we analyze two other transformations of two iid centered normal random vectors. These transformations are similar in spirit to the transformation considered by Pillai and Meng (2016). One transformation involves absolute values: ∑j=1nwjXj/|Yj|. The second involves randomly stopped Brownian motions: ∑j=1nwjXj(Yj−2), where {(X1(t),…,Xn(t)),t≥0},n>1, is a Brownian motion with positive variances; (Y1,…,Yn) is a centered normal random vector with the same law as (X1(1),…,Xn(1)) and independent of it; and X(Y−2) is the value of the Brownian motion X(t) evaluated at the random time t=Y−2. All three transformations result in a Cauchy distribution if the covariance matrix of the normal components is diagonal, or if all the correlations implied by the covariance matrix equal 1. However, while the transformation that Pillai and Meng considered produces a Cauchy distribution regardless of the normal covariance matrix, the transformations we consider here do not always produce a Cauchy distribution. The correlations between jointly normal random variables are not always overwhelmed by the heaviness of the marginal tails. The mysteries of the connections between normal and Cauchy laws remain to be understood.

Simulation of surface flashover probability distribution with gas-solid charge transport/multiplication coupling of epoxy resin

<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.

Nonlinear Random Vibrations Using Second-Order Adjoint and Projected Differentiation Methods

Abstract This paper proposes a new computationally efficient methodology for random vibrations of nonlinear vibratory systems using a time-dependent second-order adjoint variable (AV2) method and a second-order projected differentiation (PD2) method. The proposed approach is called AV2–PD2. The vibratory system can be excited by stationary Gaussian or non-Gaussian random processes following the traditional translation process model. A Karhunen–Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. A second-order adjoint approach is used to obtain the required first- and second-order output derivatives accurately by solving as many sets of equations of motion (EOMs) as the number of KL random variables. These derivatives are used to compute the marginal cumulative distribution function (CDF) of the output process with second-order accuracy. Then, a second-order projected differentiation method calculates the autocorrelation function of each output process with second-order accuracy, at an additional cost of solving as many sets of EOMs as the number of outputs of interest, independently of the time horizon (simulation time). The total number of solutions of the EOM scales linearly with the number of input KL random variables and the number of output processes. The efficiency and accuracy of the proposed approach are demonstrated using a nonlinear Duffing oscillator problem under a quadratic random excitation and a nonlinear half-car suspension example.