Moment Theorem Research Articles

Bulk universality for complex non-Hermitian matrices with independent and identically distributed entries

AbstractWe consider $$N\times N$$ N × N matrices with complex entries that are perturbed by a complex Gaussian matrix with small variance. We prove that if the unperturbed matrix satisfies certain local laws then the bulk correlation functions are universal in the large N limit. Assuming the entries are independent and identically distributed (iid) with a common distribution that has finite moments, the Gaussian component is removed by the four moment theorem of Tao and Vu.

Improved Bound of Four Moment Theorem and Its Application to Orthogonal Polynomials Associated with Laws

In the case where the square of an eigenfunction F with respect to an eigenvalue of Markov generator L can be expressed as a sum of eigenfunctions, we find the largest number excluding zero among the eigenvalues in the terms of the sum. Using this number, we obtain an improved bound of the fourth moment theorem for Markov diffusion generators. To see how this number depends on an improved bound, we give some examples of eigenfunctions of the diffusion generators L such as Ornstein–Uhlenbeck, Jacobi, and Romanovski–Routh.

Mechanical model and numerical simulation of oblique penetration of projectile into foamed aluminum laminated target

To accurately predict the trajectory and attitude of the projectile obliquely penetrating the laminated aluminum foam target, the penetration resistance function which conclude normal stress, shear stress and friction is established in this paper. Then, we establish the differential equation of the projectile motion combined with the momentum theorem and the momentum moment theorem. Finally, we use finite element software Ansys/Ls-Dyna to simulate the oblique penetration, and the attitude deflection under different initial velocity or different initial attitude angle is emphatically analyzed. The results show that there is a peak of attitude deflection at the beginning of oblique penetration, and the peak is little correlated with initial velocity, but is positively correlated with initial attitude angle. The projectile will deflect three times during oblique penetration, and the variation is “increase-decrease-increase”. When the projectile finally through the target, the value of attitude deflection is negatively correlated with initial velocity and positively correlated with initial attitude angle.

Design and Experimental Study on the Torque Balancing Mechanism of a Satellite-Borne Two-Axis Rotary Table

With the rapid development of science and technology, high-resolution remote sensing cameras are now widely used in various fields. At the beginning of camera attitude adjustment, residual torque due to state changes can affect platform stability and lead to the degradation of imaging quality. This paper analyzes the effect of external disturbances on the attitude of the satellite platform according to the Newton–Euler method. In order to effectively realize the self-balancing of the torque of the remote sensing camera rotary table and eliminate the influence of the residual torque on the stability of the satellite platform, this study designs a torque balancing mechanism for the two-axis rotary table of a remote sensing camera based on the first-generation balancing mechanism. Firstly, this paper provides a detailed analysis of the mechanism equilibrium principle from the theoretical point of view based on the theories related to momentum moment theorem and momentum moment conservation law. Then, the dynamics model of the torque balancing mechanism is built, and the dynamics simulation analysis is carried out in this paper. The analysis results show that compared with the first-generation torque balancing mechanism, the residual torque of the second-generation torque balancing mechanism is reduced by 66.67%, the peak value of the residual torque is reduced by 57.55%, the mass of the balancing flywheel is reduced by 74.14%, and the torque balancing time is reduced by 42.86%. Finally, two torque balancing mechanism prototypes were fabricated at equal scale for test verification in this paper. The test results show that compared with the first-generation torque balancing mechanism, the residual torque of the second-generation test prototype is reduced by 40%, the peak of the residual torque is reduced by 25%, the mass of the balancing flywheel is reduced by 60.34%, and the torque balancing time is reduced by 51.06%. There are some differences between the simulation analysis and the experimental results, but the overall trend is consistent with the theory. Through theoretical derivation, simulation analysis and experimental verification, the correctness and feasibility of the proposed second-generation torque balancing mechanism are fully confirmed, which has certain reference significance and engineering application value for the torque self-balancing scheme of rotary tables.

Dynamic Analysis of Underwater Torpedo during Straight-Line Navigation

Torpedoes play an irreplaceable role in naval warfare; therefore, it is significant to study the dynamic response of the direct navigation of torpedoes. In order to study the dynamic response of torpedoes under different Munk moment coefficients, the dynamic equation of torpedoes is established based on the momentum theorem and the momentum moment theorem. The linear motion mathematical model of torpedoes is obtained. The relationship between the torpedo and the Munk moment coefficient is derived. The straight-line motion model of the torpedo under different Munk moments is established, and the dynamic properties of the space motion of the torpedo are analyzed. It is found that the Munk moment coefficient increase will lead to an increase in the deflection of the torpedo’s direct motion on each degree of freedom, and the Munk moment coefficient is related to the additional mass matrix. During the design of the torpedo, the added mass should be reduced by changing the shape of the torpedo as much as possible so as to reduce the pitch moment, yaw, and roll moments of the torpedo.

Vibration and stability analysis of a spinning shaft with arbitrary boundaries subjected to partial load

This paper deals with the flexural vibration and dynamic stability of a spinning shaft subjected to partial load. Based on the theorem of moment of momentum and the theorem of motion of center of mass, the governing equation of motion for the spinning shaft system is obtained. Four kinds of boundary conditions including pinned-pinned, clamped-clamped, free-free, and elastic supports are considered. The exact whirl frequency equation is derived analytically for different boundary conditions. The obtained model is confirmed by comparing with the results reported in the available literature, and excellent agreement is presented. A numerical investigation is carried out to examine the effects of support stiffness, mass ratio, eccentric distance, and slenderness ratio on the whirl characteristics and stability of the system in terms of the obtained model. The results show that the whirl frequency and critical spinning speed of the system are dependent on these parameters, and the stability is mainly governed by the mass ratio and eccentric distance.

Prescribed Performance Adaptive Balance Control for Reaction Wheel-Based Inverted Pendulum-Type Cubli Rovers in Asteroid

This paper investigates the issue of balance control for reaction-wheeled inverted pendulum-type Cubli Rovers on asteroids, and an adaptive control scheme is proposed via the prescribed performance control technique. The main feature lies in the fact that the transient behavior is satisfied which is required critically in the environment of asteroids. The attitude model of reaction-wheeled inverted pendulum-type Cubli Rovers is first constructed by virtue of the momentum moment theorem and Eulerian kinematics. Based on that, the gravitational field in the asteroid is described and the avoiding jumping condition is analyzed. Then, an adaptive prescribed performance control (APPC) method is proposed to obtain the fine tracking performance of the equilibrium error such that the inverted pendulum-type Cubli Rovers achieve the self-balancing motion. The proposed method is capable of ensuring the tracking errors inside the preset boundary functions, and the asymptotic stability of all states in the closed-loop system is guaranteed via the Lyapunov stability theory. The simulation and comparison results on the environment of asteroids verify the effectiveness and superiority of the presented control law.

Statistical properties of partially coherent beams transmitted through nonlinear Kerr thin layers

Investigation of the propagation of partially coherent beams through nonlinear media is very difficult because of the need to deal with the fourth- or higher-order correlation function. Here a simple theoretical model is proposed to derive analytical results on the transmission of partially coherent beams through nonlinear Kerr thin layers. Considering only the self-phase modulation effect and neglecting the diffraction effect, the exact probability density function of the transmitted field at arbitrary two points has been derived and found to be not Gaussian. Statistical properties of the transmitted Gaussian Schell-model beams are studied in detail. The moment theorem is no longer valid so that the fourth-order correlation function cannot be decomposed as the sum of products of the second-order correlation functions. Numerical results are presented to see the effects of nonlinearity on the deviation from Gaussian statistics.

Moment estimator for an AR(1) model driven by a long memory Gaussian noise

In this paper, we consider an inference problem for the first order autoregressive process driven by a given long memory stationary Gaussian noise. Suppose that the covariance function of the noise is positive and can be expressed as |k|2H−2 times a positive function slowly varying at infinity. The fractional Gaussian noise and the fractional ARIMA(0, d, 0) model with d∈(0,12) and some others Gaussian noise are special examples that satisfy this assumption. We propose a moment estimator and prove the strong consistency, the asymptotic normality and the almost sure central limit theorem. Moreover, we give the upper Berry–Esséen bound by means of Fourth moment theorem.

Stein factors for variance-gamma approximation in the Wasserstein and Kolmogorov distances

We obtain new bounds for the solution of the variance-gamma (VG) Stein equation that are of the correct form for approximations in terms of the Wasserstein and Kolmogorov metrics. These bounds hold for all parameters values of the four parameter VG class. As an application we obtain explicit Wasserstein and Kolmogorov distance error bounds in a six moment theorem for VG approximation of double Wiener-Itô integrals.

Fourth Cumulant Bound of Multivariate Normal Approximation on General Functionals of Gaussian Fields

We develop a technique for obtaining the fourth moment bound on the normal approximation of F, where F is an Rd-valued random vector whose components are functionals of Gaussian fields. This study transcends the case of vectors of multiple stochastic integrals, which has been the subject of research so far. We perform this task by investigating the relationship between the expectations of two operators Γ and Γ*. Here, the operator Γ was introduced in Noreddine and Nourdin (2011) [On the Gaussian approximation of vector-valued multiple integrals. J. Multi. Anal.], and Γ* is a muilti-dimensional version of the operator used in Kim and Park (2018) [An Edgeworth expansion for functionals of Gaussian fields and its applications, stoch. proc. their Appl.]. In the specific case where F is a random variable belonging to the vector-valued multiple integrals, the conditions in the general case of F for the fourth moment bound are naturally satisfied and our method yields a better estimate than that obtained by the previous methods. In the case of d=1, the method developed here shows that, even in the case of general functionals of Gaussian fields, the fourth moment theorem holds without conditions for the multi-dimensional case.

Optimal Variance–Gamma approximation on the second Wiener chaos

In this paper, we consider a target random variable Y∼VGc(r,θ,σ) distributed according to a centered Variance–Gamma distribution. For a generic random element F=I2(f) in the second Wiener chaos with E[F2]=E[Y2] we establish a non-asymptotic optimal bound on the distance between F and Y in terms of the maximum difference of the first six cumulants. This six moment theorem complements and in some sense extends the celebrated optimal fourth moment theorem of I. Nourdin & G. Peccati for normal approximation. The main body of our analysis constitutes a splitting technique for test functions in the Banach space of Lipschitz functions relying on the compactness of the Stein operator. The recent developments around Stein method for Variance–Gamma approximation by R. Gaunt play a significant role in our study. As an application we consider the generalized Rosenblatt process at the extreme critical exponent, first studied by S. Bai & M. Taqqu.

Normal approximation when a chaos grade is greater than two

We derive an upper bound on a probabilistic distance for a normal approximation when the chaos grade of an eigenfunction of Markov diffusion generator L is greater than 2. When a chaos grade is strictly greater than 2, the upper bound, given by Bourguin et al. (2019), does not guarantee that Fn converges in distribution to a standard Gaussian distribution even when the fourth cumulant of Fn converges to 0. This means that the fourth moment theorem, discovered by Nualart and Peccati (2005), does not work. In this paper, we develop a new technique to obtain an upper bound for which the fourth moment theorem works when a chaos grade is strictly greater than 2.

New error bounds in multivariate normal approximations via exchangeable pairs with applications to Wishart matrices and fourth moment theorems

We extend Stein’s celebrated Wasserstein bound for normal approximation via exchangeable pairs to the multi-dimensional setting. As an intermediate step, we exploit the symmetry of exchangeable pairs to obtain an error bound for smooth test functions. We also obtain a continuous version of the multi-dimensional Wasserstein bound in terms of fourth moments. We apply the main results to multivariate normal approximations to Wishart matrices of size n and degree d, where we obtain the optimal convergence rate n3/d under only moment assumptions, and to degenerate U-statistics and Poisson functionals, where we strengthen a few of the fourth moment bounds in the literature on the Wasserstein distance.

Generalizations of the fourth moment theorem

Generalizations of the fourth moment theorem

Partial generalized four moment theorem revisited

This is a complementary proof of partial generalized 4 moment theorem (PG4MT) mentioned and described in “Generalized Four Moment Theorem (G4MT) and its Application to CLT for Spiked Eigenvalues of High-dimensional Covariance Matrices”. Since the G4MT proposed in that paper requires both the matrices X and Y satisfying the assumption maxt,s|uts|2E{|x11|4I(|x11|<n)−μ}→0 with the same μ which maybe restrictive in real applications, we proposed a new G4MT, called PG4MT, without proof. After the manuscript posed in ArXiv, the authors received high interests in the proof of PG4MT through private communications and find the PG4MT more general than G4MT, it is necessary to give a detailed proof of it. Moreover, it is found that the PG4MT derives a CLT of spiked eigenvalues of sample covariance matrices which covers the work in Bai and Yao (J. Multivariate Anal. 106 (2012) 167–177) as a special case.

Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials

In this paper, we consider a quantitative fourth moment theorem for functions (random variables) defined on the Markov tripleE,μ,Γ, whereμis a probability measure andΓis the carré du champ operator. A new technique is developed to derive the fourth moment bound in a normal approximation on the random variable of a general Markov diffusion generator, not necessarily belonging to a fixed eigenspace, while previous works deal with only random variables to belong to a fixed eigenspace. As this technique will be applied to the works studied by Bourguin et al. (2019), we obtain the new result in the case where the chaos grade of an eigenfunction of Markov diffusion generator is greater than two. Also, we introduce the chaos grade of a new notion, called the lower chaos grade, to find a better estimate than the previous one.

Least-squares estimation for the Vasicek model driven by the complex fractional Brownian motion

In this paper, we study the least-squares estimation problem for the Vasicek model driven by the complex fractional Brownian motion where is a two-dimensional fractional Brownian motion with Hurst parameter We obtain the strong consistency and asymptotic normality of and using the Garsia–Rodemich–Rumsey inequality and complex fourth moment theorems.

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

AbstractWe establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.

Normal approximation and fourth moment theorems for monochromatic triangles

AbstractGiven a graph sequence denote by T3(Gn) the number of monochromatic triangles in a uniformly random coloring of the vertices of Gn with colors. In this paper we prove a central limit theorem (CLT) for T3(Gn) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well‐known fourth‐moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth‐moment condition characterizes the limiting normal distribution of T3(Gn), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].