Sensitivity of the Hermite rank

The Hermite rank appears in limit theorems involving long memory. We show that a Hermite rank higher than one is unstable when the data is slightly perturbed by transformations such as shift and scaling. We carry out a “near higher order rank analysis” to illustrate how the limit theorems are affected by a shift perturbation that is decreasing in size. We also consider the case where the deterministic shift is replaced by centering with respect to the sample mean. The paper is a companion of Bai and Taqqu (2017) which discusses the instability of the Hermite rank in the statistical context.     

Convergence of Hermite interpolatory operators

Divergence-free wavelets play important roles in both partial differential equations and fluid mechanics. Many constructions of those wavelets depend usually on Hermite splines. We study several types of convergence of the related Hermite interpolatory operators in this paper. More precisely, the uniform convergence is firstly discussed in the second part; then, the third section provides the convergence in the Donoho’s sense. Based on these results, the last two parts are devoted to show the convergence in some Besov spaces, which concludes the completeness of Bittner and Urban’s expansions.     

Polar spaces of arbitrary rank

The incidence systems which satisfy the main axiom of Buekenhout and Shult, other than those related to generalized quadrangles, now have a satisfactory classification. Here the ones of infinite rank with no lines of cardinality 2 are characterized. They are the natural analogues of those of finite rank, and the nondegenerate ones arise from polarities or pseudoquadratic forms in the usual way. The treatment quotes some fundamental theorems of Tits, who handled the finite rank case, but is otherwise self-contained. A full exposition of the relevant concepts, with a new development of some of the elementary theory, is also given.Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthday     

Convergence of integrated superpositions of Ornstein-Uhlenbeck processes to fractional Brownian motion

Superpositions of Ornstein-Uhlenbeck processes provide convenient ways to build stationary processes with given marginal distributions and long range dependence. After reviewing some of the basic features, we present several examples of processes with non-Gaussian marginal distributions. Our main results concern asymptotic properties of sums and partial sums of these processes and their polynomial functions. Further, we discuss some applications to estimation.     

Analysis and Convergence of Hermite Subdivision Schemes

Foundations of Computational Mathematics - Hermite interpolation property is desired in applied and computational mathematics. Hermite and vector subdivision schemes are of interest in CAGD for...     

On hypercomplexifying real forms of arbitrary rank

For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary Clifford algebras, followed by quaternionic vectors as a special case. All results are shown to reduce to the established method of complexifying vector fields. For simplicity, differential forms are used rather than vector notation.     

Generalized killing tensors of arbitrary rank and order

We define Killing tensors and conformal Killing tensors of arbitrary rank and order which generalize in a natural way the notion of a Killing vector. We explicitly derive the corresponding tensors for a flat de Sitter space of dimension p+q=m,m 4, which permits us to calculate complete sets of symmetry operators of arbitrary order n for a scalar wave equation with m independent parameters.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 786–795, June, 1991.     

Regularization and integral representations of Hermite processes

It is known that Hermite processes have a finite-time interval representation. For fractional Brownian motion, the representation has been well known and plays a fundamental role in developing stochastic calculus for the process. For the Rosenblatt process, the finite-time interval representation was originally established by using cumulants. The representation was extended to general Hermite processes through the convergence of suitable partial sum processes. We provide here an alternative and different proof for the finite-time interval representation of Hermite processes. The approach is based on regularization of Hermite processes and the fractional Gaussian noises underlying them, and does not use cumulants nor convergence of partial sums.     

Convergence of diffusion processes

For stochastic diffusion equations with coefficients depending on a parameter, necessary and sufficient conditions of the weak convergence of solutions to the solution of a stochastic diffusion equation are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 284–289, February, 1992.     

实轴上Hermite插值算子的收敛性

The present paper investigates the convergence of Hermite interpolation operators on the real line.The main result is: Given 0 <δ0 < 1/2,0 < ε0 < 1.Let f ∈ C(-∞,∞) satisfy |yk| = O(e(1/2-δ0)x2k) and |f(x)| = O(e(1-ε0)x2).Then for any given point x ∈R,we have limn→∞ Hn(f,x) = f(x).     

实轴上Hermite插值算子的收敛性

The present paper investigates the convergence of Hermite interpolation operators on the real line. The main result is： Given 0 〈 δo 〈 1/2, 0 〈 εo 〈 1. Let f ∈ C（-∞,∞） satisfy ｜y｜= O（e^（1/2-δo）xk^2,） and ｜f（x）｜t= O（e^（1-εo ）x2^）. Then for any given point x ∈ R, we have limn→Hn,（f, x） = f（x）.     

Convergence of Bellman-Harris branching processes

A limit theorem regarding the convergence of a sequence of normalized Bellman-Harris processes to an Jirina process (a branching process with a continuous phase space) is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 827–832, June, 1990.     

Convergence of projected Landweber iteration for matrix rank minimization

In this paper, we study the performance of the projected Landweber iteration (PLW) for the general low rank matrix recovery. The PLW was first proposed by Zhang and Chen (2010) [43] based on the sparse recovery algorithm APG (Daubechies et al., 2008) [14] in the matrix completion setting, and numerical experiments have been given to show its efficiency (Zhang and Chen, 2010) [43]. In this paper, we focus on providing a convergence rate analysis of the PLW in the general setting of low rank matrix recovery with the affine transform having the matrix restricted isometry property. We show robustness of the algorithm to noise with a strong geometric convergence rate even for noisy measurements provided that the affine transform satisfies a matrix restricted isometry property condition.