Polynomial Moments Research Articles

Polynomial moments with a weighted zeta square measure on the critical line

Polynomial moments with a weighted zeta square measure on the critical line

Sparse precision matrix estimation under lower polynomial moment assumption

Precision matrix (inverse covariance matrix) estimation is a rising challenge in contemporary applications while dealing with high‐dimensional data. This paper focuses on large‐scale precision matrix of the random vector that only has lower polynomial moments. We mainly investigate upper bounds of the proposed estimator under the spectral norm in terms of the probability and mean estimation respectively. It is shown that the data‐driven estimator is fully adaptive and achieves the same optimal convergence order as under Gaussian assumption on the data. Simulation studies further support our theoretical claims.

Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths

We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification are important because they are relatively easy to simulate. One does not have to care about the distribution of their jumps which is always difficult to find. Among those processes with the continuous path are the Ornstein–Uhlenbeck process, the Gamma process, the process with Arcsin or Wigner margins and the Theta functions as the transition densities and others. We give a simple criterion for the stationary process to have a continuous path modification expressed in terms of skewness and excess kurtosis of the marginal distribution.

A Stieltjes Algorithm for Generating Multivariate Orthogonal Polynomials

Orthogonal polynomials of several variables have a vector-valued three-term recurrence relation, much like the corresponding one-dimensional relation. This relation requires only knowledge of certain recurrence matrices, and allows simple and stable evaluation of multivariate orthogonal polynomials. In the univariate case, various algorithms can stably and accurately evaluate the recurrence coefficients given the ability to compute polynomial moments, but it is difficult to identify analogous procedures in multiple dimensions. We present a new Multivariate Stieltjes (MS) algorithm that fills this gap in the multivariate case, allowing computation of recurrence matrices assuming moments are available. The algorithm is essentially explicit in two and three dimensions, but requires the numerical solution to a nonconvex problem in more than three dimensions. Compared to direct Gram–Schmidt-type orthogonalization, we demonstrate on several examples in up to three dimensions that the MS algorithm is far more stable, and allows accurate computation of orthogonal bases in the multivariate setting, in contrast to direct orthogonalization approaches.

Uncertainty analysis of MSD crack propagation based on polynomial chaos expansion

Widespread fatigue damage (WFD) is one of the main damages in aging aircraft structures, and the prediction of multi-site damage (MSD) crack propagation plays an important role in the safety assessment of aging aircraft. However, the existing multi-crack propagation stochastic models need to calculate multiple integrals, and the calculation efficiency is low. Based on the Polynomial Chaos Expansion (PCE) method, this paper proposes a new uncertainty analysis method to effectively solve the problem of multi-crack mutual interference. The rivet area of aircraft skin mostly appears randomly in the form of short cracks. Therefore, 30 groups of multi-crack fatigue propagation experiments are carried out to obtain the data of multi-cracks at different stages. On the basis of the existing multi-crack mutual interference model, a new semi-analytical multi-crack propagation model is designed by considering the uncertainty of multi-crack initiation and the interaction factors between cracks. The random variables are substituted into the improved multi-crack propagation model, and the Hermite polynomials are introduced and expanded into orthogonal polynomial series, and the time-varying polynomial coefficients, moments and probability densities are solved by stochastic regression method. The accuracy and effectiveness of the method are verified by comparing with the experimental and Monte Carlo methods.

For which functions are 𝑓(𝑋_{𝑡})-𝔼𝕗(𝕏_{𝕥}) and 𝕘(𝕏_{𝕥})/𝔼𝕘(𝕏_{𝕥}) martingales?

Let X = ( X t ) t ≥ 0 X=(X_t)_{t\geq 0} be a one-dimensional Lévy process such that each X t X_t has a C b 1 C^1_b -density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions f : R → R f\colon \mathbb {R}\to \mathbb {R} , and exponentially bounded functions g : R → ( 0 , ∞ ) g\colon \mathbb {R}\to (0,\infty ) , such that f ( X t ) − E f ( X t ) f(X_t)-\mathbb {E} f(X_t) , resp. g ( X t ) / E g ( X t ) g(X_t)/\mathbb {E} g(X_t) , are martingales.

On smooth change-point location estimation for Poisson Processes

We are interested in estimating the location of what we call “smooth change-point” from n independent observations of an inhomogeneous Poisson process. The smooth change-point is a transition of the intensity function of the process from one level to another which happens smoothly, but over such a small interval, that its length delta _n is considered to be decreasing to 0 as nrightarrow +infty . We show that if delta _n goes to zero slower than 1/n, our model is locally asymptotically normal (with a rather unusual rate sqrt{delta _n/n}), and the maximum likelihood and Bayesian estimators are consistent, asymptotically normal and asymptotically efficient. If, on the contrary, delta _n goes to zero faster than 1/n, our model is non-regular and behaves like a change-point model. More precisely, in this case we show that the Bayesian estimators are consistent, converge at rate 1/n, have non-Gaussian limit distributions and are asymptotically efficient. All these results are obtained using the likelihood ratio analysis method of Ibragimov and Khasminskii, which equally yields the convergence of polynomial moments of the considered estimators. However, in order to study the maximum likelihood estimator in the case where delta _n goes to zero faster than 1/n, this method cannot be applied using the usual topologies of convergence in functional spaces. So, this study should go through the use of an alternative topology and will be considered in a future work.

The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise

This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump Lévy noise of small amplitude ε>0, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes Xε turns out to be exponentially ergodic with equilibrium distribution με in total variation distance which extends a result from [60] to arbitrary polynomial moments. The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index α>3 2. That is to say, in this setting we identify a deterministic time scale tεcut satisfying tεcut→∞, as ε→0, and a respective time window, tεcut±o(tεcut), during which the total variation distance between the current state and its equilibrium με essentially collapses as ε tends to zero. In addition, we extend the dynamical characterization under which the latter phenomenon can be described by the convergence of such distance to a unique profile function first established in [9] to the Lévy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric α-stable noise for α>3 2. The proof techniques differ completely from the Gaussian case due to the absence of a respective Girsanov transform which couples the nonlinear equation and the linear approximation asymptotically even for short times.

Parameter estimation for orthogonal polynomial moments

AbstractProny's method recovers a Dirac ensemble μ = Σi=1M aiδxi with given sparsity level by separating the reconstruction of the unknown coefficients and positions. While it recovers the positions by examing the so called Prony polynomial, its reconstruction is heavily based on solving linear systems, which can be constructed by given monomial moments.The intention of this article is to propose a Prony‐like method to recover the Dirac ensemble with an upper bounded sparsity level from orthogonal polynomial moments.

Local asymptotic properties for Cox‐Ingersoll‐Ross process with discrete observations

AbstractIn this paper, we consider a one‐dimensional Cox‐Ingersoll‐Ross (CIR) process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. To obtain these results, we use the Malliavin calculus techniques developed recently for CIR process together with the estimation for positive and negative polynomial moments of the CIR process. In this study, we require the same conditions of high frequency and infinite horizon as in the case of ergodic diffusions with globally Lipschitz coefficients studied earlier in the literature. However, in the non‐ergodic cases, additional assumptions on the decreasing rate are required due to the fact that the square root diffusion coefficient of the CIR process is not regular enough.

Renormalons in integrated spectral function moments and αs extractions

Precise extractions of $\alpha_s$ from $\tau\to {\rm (hadrons)}+\nu_\tau$ and from $e^+e^-\to {\rm (hadrons)}$ below the charm threshold rely on finite energy sum rules (FESRs) where the experimental side is given by integrated spectral function moments. Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-$\beta_0$ limit and in full QCD. In large-$\beta_0$, we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. The leading IR singularity is particularly prominent and is behind the fate of moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. The conclusions can be extended to QCD through a convenient scheme transformation to the $C$-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-$\beta_0$. Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-$Q^2$ $\alpha_s$ determinations.

Decay estimates for large velocities in the Boltzmann equation without cutoff

We consider solutions f=f(t,x,v) to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions x∈𝕋 d , for hard and moderately soft potentials without the angular cutoff assumption, and under the a priori assumption that the main hydrodynamic fields, namely the local mass ∫fdv and local energy ∫f|v| 2 dv and local entropy ∫flnfdv, are controlled along time. We establish quantitative estimates of propagation in time of “pointwise polynomial moments”, i.e., sup x,v f(t,x,v)(1+|v|) q , q≥0. In the case of hard potentials, we also prove appearance of these moments for all q≥0. In the case of moderately soft potentials, we prove the appearance of low-order pointwise moments. All these conditional bounds are uniform as t goes to +∞, conditionally to the bounds on the hydrodynamic fields being uniform.

On Existence and Uniqueness to Homogeneous Boltzmann Flows of Monatomic Gas Mixtures

We solve the Cauchy problem for the full non-linear homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions in three dimensions. More precisely, we show existence and uniqueness of the vector value solution by means of an existence theorem for ODE systems in Banach spaces under the transition probability rates assumption corresponding to hard potentials rates in the interval $(0,1]$, with an angular section modeled by an integrable function of the angular transition rates modeling binary scattering effects. The initial data for the vector valued solutions needs to be a vector of non-negative measures with finite total number density, momentum and strictly positive energy, as well as to have a finite $L^1_{k_*}(\mathbb{R}^3)$-integrability property corresponding to a sum across each species of $k_*$-polynomial weighted norms depending on the corresponding mass fraction parameter for each species as much as on the intermolecular potential rates, referred as to the scalar polynomial moment of order $k_*$. The existence and uniqueness rigorous results rely on a new angular averaging lemma adjusted to vector values solution that yield a Povzner estimate with constants that decay with the order of the corresponding dimensionless scalar polynomial moment. In addition, such initial data yields global generation of such scalar polynomial moments at any order as well as their summability of moments to obtain estimates for corresponding scalar exponentially decaying high energy tails, referred as to scalar exponential moments associated to the system solution. Such scalar polynomial and exponential moments propagate as well.

High-dimensional Linear Regression for Dependent Data with Applications to Nowcasting

Recent research has focused on $\ell_1$ penalized least squares (Lasso) estimators for high-dimensional linear regressions in which the number of covariates $p$ is considerably larger than the sample size $n$. However, few studies have examined the properties of the estimators when the errors and/or the covariates are serially dependent. In this study, we investigate the theoretical properties of the Lasso estimator for a linear regression with a random design and weak sparsity under serially dependent and/or nonsubGaussian errors and covariates. In contrast to the traditional case, in which the errors are independent and identically distributed and have finite exponential moments, we show that $p$ can be at most a power of $n$ if the errors have only finite polynomial moments. In addition, the rate of convergence becomes slower owing to the serial dependence in the errors and the covariates. We also consider the sign consistency of the model selection using the Lasso estimator when there are serial correlations in the errors or the covariates, or both. Adopting the framework of a functional dependence measure, we describe how the rates of convergence and the selection consistency of the estimators depend on the dependence measures and moment conditions of the errors and the covariates. Simulation results show that a Lasso regression can be significantly more powerful than a mixed-frequency data sampling regression (MIDAS) and a Dantzig selector in the presence of irrelevant variables. We apply the results obtained for the Lasso method to nowcasting with mixed-frequency data, in which serially correlated errors and a large number of covariates are common. The empirical results show that the Lasso procedure outperforms the MIDAS regression and the autoregressive model with exogenous variables in terms of both forecasting and nowcasting.

Chemical Kinetics Roots and Methods to Obtain the Probability Distribution Function Evolution of Reactants and Products in Chemical Networks Governed by a Master Equation.

In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied.

Affine Stochastic Volatility Models: Supplementary Material

Affine jump diffusion models in general and affine stochastic volatility models in particular are important modeling tools in finance. Their popularity resides in their exibility coupled with their analytical tractability, especially with respect to characteristic functions and polynomial moments. Within a generic affine jump diffusion model and a generic stochastic volatility model, nested in the former as a special case, this paper collects explicit expressions for various characteristic functions and polynomial moments.

单生过程的研究进展

The single birth process, as a natural extension of birth and death processwhich is the simplest $Q$-process, has its own origins in research background. On one hand, the single birth processis nearly the largest class for which the explicit criteria on classical problems can be expected. Thus itbecomes a fundamental comparison tool in studying complicated processes, such as infinite-dimensional reaction-diffusion processes.On the other hand, the single birth process is usually non-symmetric and hence is regarded as a representative of the non-symmetric processes.For non-symmetric processes, in contrast to symmetric ones, our knowledge is very limited, except for single birth processes to which many results arerelatively completed. In this survey paper, we present some criteria on classical problems of single birth processes, including uniqueness, recurrence, ergodicity and strong ergodicity, as well as representation of return(or extinction) probability and stationary distribution. Recently explicit representation of solution of Poissons equation has been obtained. Based on this tool,a unified treatment of various problems for general single birth processes is presented. To illustrate this approach we deal with uniqueness, recurrence and return probability as examples. Polynomial moments and exponential onesof return time as well as the distribution of explosion time are obtained similarly. Moreover, we present an explicit and sufficient condition for exponential ergodicity of single birth processes. At last, we review someconcrete applications of single birth processes in the study of particle systems and other models.

SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations

We present the design, convergence analysis and numerical investigations of the nonconforming virtual element method with Streamline Upwind/Petrov–Galerkin (VEM-SUPG) stabilization for the numerical resolution of convection–diffusion–reaction problems in the convective-dominated regime.According to the virtual discretization approach, the bilinear form is split as the sum of a consistency and a stability term. The consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form (including the SUPG stabilization term). Polynomial projections can be computed exactly from the degrees of freedom. The stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients.The nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface. Optimal convergence properties of the method are proved in a suitable norm, which includes contribution from the advective stabilization terms. Experimental results confirm the theoretical convergence rates.

Well‐posedness of Bayesian inverse problems in quasi‐Banach spaces with stable priors

AbstractThe Bayesian perspective on inverse problems has attracted much mathematical attention in recent years. Particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an infinite‐dimensional space, a typical example being a scalar or tensor field coupled to some observed data via an ODE or PDE. This article gives an introduction to the framework of well‐posed BIPs in infinite‐dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559, 2010) and others. This framework has the advantage of ensuring uniformly well‐posed inference problems independently of the finite‐dimensional discretisation used for numerical solution. Recently, this framework has been extended to the case of a heavy‐tailed prior measure in the family of stable distributions, such as an infinite‐dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loève expansion for square‐integrable random variables can be used to sample such measures on quasi‐Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger and total variation metrics upon perturbations of the misfit function and observed data. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Delamination Identification in Stiffened Composite Panels Using Surface Strain Data

The structural integrity of a composite structure can be greatly compromised by damage inside the component. Invisible damage, e.g. caused by low-speed impact, can significantly reduces composite components capability to efficiently carry loads. In the present study, an innovative approach of strain-based delamination identification and localization is investigated, based on the efficient processing of full-field surface strain measurements. Surface strain data, potentially derived by full-field optical methodologies are used in the assessment of the delamination pattern, through strain field perturbations caused due to damage evolution. Relations between delamination damage and surface strain field disturbances are established by exploiting data decomposition methods using Zernike polynomial moments. The methodology is successfully demonstrated in the case of a stiffened composite panel.